# Iterated power intersections of ideals in rings of iterated differential
polynomials II

Pavel Příhoda  Charles University, Faculty of Mathematics and Physics,
Department of Algebra, Sokolovská 83, 18675 Prague 8, Czech Republic
<EMAIL_ADDRESS>
###### Abstract.

This paper is a continuation of [6] where iterated intersections of powers of
ideals were studied in rings of iterated differential polynomials. We present
a method which can be used to show that for every proper ideal $I$ of a
suitable ring of iterated differential polynomials almost all iterated
intersections of powers of $I$ have to be zero. The setback of the method is
that it works only if the derivations used in the construction of the ring of
iterated differential polynomials satisfy additional assumptions. On the other
hand, it can be applied in cases which were not covered by [6], for example
for a universal enveloping algebra of a completely solvable Lie algebra over a
field of positive characteristic.

###### 2010 Mathematics Subject Classification:

16D25,16D40,16S30

This research was supported by Czech Science Foundation grant GA ČR 20-13778S

##  1\. Introduction

Let $I$ be an ideal of a ring $R$ and let $I(1):=\cap_{n\in\mathbb{N}}I^{n}$
denote the intersection of all its powers. Let us define inductively
$I(m):=(I(m-1))(1)$ for every $2\leq m\in\mathbb{N}$. The ideal $I(m)$ is
called the $m$-iterated intersection of powers of $I$. This short note gives
some examples of rings having the following property: For any proper ideal $I$
of $R$ there exists $m\in\mathbb{N}$ such that $I(m)=0$.

This property was studied for rings of iterated differential polynomials in
[6]. Recall that if $R$ is a ring and $\delta\colon R\to R$ is a derivation on
$R$, i.e., an additive map satisfying $\delta(rs)=\delta(r)s+r\delta(s)$ for
every $r,s\in R$, the ring of differential polynomials $R[x,\delta]$ is a free
left $R$-module with basis $1,x,x^{2},\dots$ with a multiplication given by
the rule $xr-rx=\delta(r),r\in R$ (see for example [3, Chapter 2] for details
of this construction). Iterating this construction $n$-times we get a ring
$S:=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ of iterated differential
polynomials over $R$. For example, if $R_{i}:=R[x_{1},\dots,x_{i}]$ is the
subring of $S$ generated by $R$ and $x_{1},\dots,x_{i}$, then $\delta_{i+1}$
is a derivation on $R_{i}$ and $R_{i+1}:=R[x_{1},\dots,x_{i+1}]$ is isomorphic
to $R_{i}[x,\delta_{i+1}]$.

By [6, Theorem 3.4], if $R$ is a commutative noetherian domain which is a
$\mathbb{Q}$-algebra and $I$ is a proper ideal of
$R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ a ring of iterated differential
polynomials over $R$, then $I(n+1)=0$.

A natural source of examples of rings of differential polynomials are
universal enveloping algebras of solvable Lie algebras of finite dimension. By
[6, Proposition 4.1], if $L$ is a solvable Lie algebra of finite dimension
over a field of characteristic $0$, then $I(2)=0$ for every proper ideal $I$
of the universal enveloping algebra of $L$.

The approach used in [6] depends on the fact that each prime ideal in the
considered ring of iterated differential polynomials is completely prime (cf.
[3, Theorem 10.23] which is a key ingredient in the proof of [6, Theorem
3.4]). In particular, this approach does not work for algebras over fields of
positive characteristic.

We offer an elementary method which can be applied to a differential
polynomial ring
$S=R[x_{1},\delta_{1},x_{2},\delta_{2},\dots,x_{n},\delta_{n}]$ even in
positive characteristic provided the derivations satisfy additional
properties. For example, assume that

  1. (a)

$\delta_{i}(R)\subseteq R$ for any $1\leq i\leq n$,

  2. (b)

for every $1\leq j<i\leq n$ there are $u,v\in R[x_{1},\dots,x_{j-1}]$ such
that $\delta_{i}(x_{j})=ux_{j}+v$,

  3. (c)

$R$ is right noetherian and there are no ideals of $R$ invariant under
$\delta_{1},\dots,\delta_{n}$ with exception of $0$ and $R$.

In Theorem 3.6 we prove that in this case every proper ideal $I$ of $S$
satisfies $I(n)=0$. A natural example of a ring of differential polynomials
satisfying $(a),(b)$ and $(c)$ is a universal enveloping algebra of a
completely solvable Lie algebra over a field.

Our main motivation to study iterated power intersections of ideals stems from
their relation with the absence of interesting non-finitely generated
projective modules.

######  Proposition 1.1.

([6, Proposition 3.7]) Let $S$ be a ring such that $S/J(S)$ is right
noetherian. Assume that for every proper ideal $I$ of $S$ there exists
$m\in\mathbb{N}$ such that $I(m)=0$. Then every projective right $S$-module
which is not finitely generated is free.

Proof. As explained in the proof of [6, Proposition 3.7], the assumption on
ideals of $S$ implies that every countably but not finitely generated
projective right $S$-module $P$ is uniformly $\aleph_{0}$-big in the sense of
Bass [2], that is, if $P/PI$ is finitely generated for some ideal $I$ of $S$,
then $I=S$. The result then follows from [2, Theorem 3.1].

##  2\. Reducing the number of variables via localization and factorization

Any ring of iterated differential polynomials
$S=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ in $n$ variables over $R$ can
be considered as a ring of iterated differential polynomials in $n-1$
variables over $R[x_{1},\delta_{1}]$, i.e.,
$S=(R[x_{1},\delta_{1}])[x_{2},\delta_{2},\dots,x_{n},\delta_{n}]$. In the
proofs of Theorem 3.5 and Theorem 3.6 we consider right localizations of $S$
with respect to right denominator sets of $R$ and $R[x_{1},\delta_{1}]$ and
factors of $S$ modulo ideals of $S$ which are generated by ideals of
$R[x_{1},\delta_{1}]$ invariant under derivations
$\delta_{2},\dots,\delta_{n}$. Although these constructions are quite
standard, in this section we recall them in detail in order to explain the
notation which is used in the proofs of the results in Section 3.

Let $R$ be a ring, $\delta\colon R\to R$ a derivation on $R$ and let
$X\subseteq R$ be a right denominator set of $R$ (for details on right
quotient rings see for example [5, Chapter 2]). By [3, Exercise 10R], $X$ is a
right denominator set of $S=R[x,\delta]$ and the right localization $SX^{-1}$
of $S$ with respect to $X$ has a structure of a differential polynomial ring.
To be more precise, let $\theta\colon R\to RX^{-1}$ be a right localization of
$R$ with respect to $X$. Then there exists a unique derivation $\partial\colon
RX^{-1}\to RX^{-1}$ on $RX^{-1}$ satisfying $\partial\theta=\theta\delta$ and
the canonical map $\varphi\colon R[x,\delta]\to RX^{-1}[x,\partial]$ which
maps $\sum_{i=0}^{d}r_{i}x^{i}$ to $\sum_{i=0}^{d}\theta(r_{i})x^{i}$ is a
right localization of $R[x,\delta]$ with respect to $X$.

Of course, this construction can be iterated: If
$S=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ is a ring of iterated
differential polynomials and $X\subseteq R$ is a right denominator set of $R$,
then $X$ is a right denominator set in $S$ and the ring $SX^{-1}$ has a
structure of a ring of iterated differential polynomials. Let $\theta\colon
R\to RX^{-1}$ be as above and suppose we have defined a ring
$RX^{-1}[x_{1},\partial_{1},\dots,x_{i},\partial_{i}]$ (i.e., we have defined
derivations $\partial_{1},\dots,\partial_{i}$) such that the canonical map

$\varphi_{i}\colon R[x_{1},\delta_{1},\dots,x_{i},\delta_{i}]\to
RX^{-1}[x_{1},\partial_{1},\dots,x_{i},\partial_{i}]$

which sends a term $rx_{1}^{e_{1}}\cdots x_{i}^{e_{i}}$ to
$\theta(r)x_{1}^{e_{1}}\cdots x_{i}^{e_{i}}$ is a right localization of
$R[x_{1},\delta_{1},\dots,x_{i},\delta_{i}]$ with respect to $X$. By the
single variable case discussed above, $X$ is a right denominator set in
$R[x_{1},\delta_{1},\dots,x_{i+1},\delta_{i+1}]$ and there exists a unique
derivation

$\partial_{i+1}\colon RX^{-1}[x_{1},\partial_{1},\dots,x_{i},\partial_{i}]\to
RX^{-1}[x_{1},\partial_{1},\dots,x_{i},\partial_{i}]$

on $RX^{-1}[x_{1},\partial_{1},\dots,x_{i},\partial_{i}]$ such that
$\varphi_{i}\delta_{i+1}=\partial_{i+1}\varphi_{i}$. Then the canonical map

$\varphi_{i+1}\colon R[x_{1},\partial_{1},\dots,x_{i+1},\partial_{i+1}]\to
RX^{-1}[x_{1},\partial_{1},\dots,x_{i+1},\partial_{i+1}]$

which sends a term $rx_{1}^{e_{1}}\cdots x_{i+1}^{e_{i+1}}$ to
$\theta(r)x_{1}^{e_{1}}\cdots x_{i+1}^{e_{i+1}}$ is a right localization of
$R[x_{1},\delta_{1},\dots,x_{i+1},\delta_{i+1}]$ with respect to $X$.

Repeating this process $n$-times, we define a ring
$RX^{-1}[x_{1},\partial_{1},\dots,x_{n},\partial_{n}]$ such that the canonical
map

$\varphi\colon S\to RX^{-1}[x_{1},\partial_{1},\dots,x_{n},\partial_{n}]$

which sends a term $rx_{1}^{e_{1}}\cdots x_{n}^{e_{n}}$ to
$\theta(r)x_{1}^{e_{1}}\cdots x_{n}^{e_{n}}$ is a right localization of $S$
with respect to $X$.

Now consider a ring of iterated differential polynomials
$S=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ and an ideal $P\subseteq R$ of
$R$ such that $\delta_{i}(P)\subseteq P$ for every $1\leq i\leq n$. Then
$PS=P[x_{1},x_{2},\dots,x_{n}]$ is the ideal of $S$ generated by $P$. The
factor ring $\overline{S}=S/PS$ can be seen as a ring of iterated differential
polynomials over $R/P$: Let $\pi\colon R\to R/P$ be the canonical projection.
Since $P$ is $\delta_{1}$-invariant, there exists a derivation
$\overline{\delta_{1}}\colon R/P\to R/P$ such that
$\pi\delta_{1}=\overline{\delta_{1}}\pi$. The canonical map $\pi_{1}\colon
R[x_{1},\delta_{1}]\to R/P[x_{1},\overline{\delta_{1}}]$ is a homomorphism of
rings with kernel $P[x_{1}]$.

Assume we have defined $\overline{\delta_{1}},\dots,\overline{\delta_{i}}$
such that the canonical map

$\pi_{i}\colon R[x_{1},\delta_{1},\dots,x_{i},\delta_{i}]\to
R/P[x_{1},\overline{\delta_{1}},\dots,x_{i},\overline{\delta_{i}}]$

which sends $rx_{1}^{e_{1}}\cdots x_{i}^{e_{i}}$ to $\pi(r)x_{1}^{e_{1}}\cdots
x_{i}^{e_{i}}$ is an onto ring homomorphism with kernel
$P[x_{1},\dots,x_{i}]$. Then $P[x_{1},\dots,x_{i}]$ is stable under
$\delta_{i+1}$. Then there exists a derivation $\overline{\delta_{i+1}}\colon
R/P[x_{1},\overline{\delta}_{1},\dots,x_{i},\overline{\delta}_{i}]\to
R/P[x_{1},\overline{\delta}_{1},\dots,x_{i},\overline{\delta}_{i}]$ such that
$\overline{\delta_{i+1}}\pi_{i}=\pi_{i}\delta_{i+1}$. Then the canonical map

$\pi_{i+1}\colon R[x_{1},\delta_{1},\dots,x_{i+1},\delta_{i+1}]\to
R/P[x_{1},\overline{\delta_{1}},\dots,x_{i+1},\overline{\delta_{i+1}}]$

is a ring homomorphism with kernel $P[x_{1},\dots,x_{i+1}]$.

After $n$ repetitions of this process we define on $S/PS$ a structure of a
ring of iterated differential polynomials.

##  3\. The results

Let us single out several easy observations we are going to use repeatedly in
the proofs.

######  Lemma 3.1.

Let $\varphi\colon R\to S$ be a homomorphism of rings, $I$ an ideal of $R$,
$J$ an ideal of $S$. If $\varphi(I)\subseteq J$ and $J(n)=0$ for some
$n\in\mathbb{N}$, then $\varphi(I(n))=0$.

Proof. If $\varphi(I)\subseteq J$, then it is easy to check by induction that
$\varphi(I(k))\subseteq J(k)$ for every $k\in\mathbb{N}$. Then the result is
immediate.

######  Lemma 3.2.

Let $R$ be a ring and let $I\subseteq R$ be its ideal satisfying $I(1)=0$. If
there exists $n\in\mathbb{N}$ such that every proper ideal $J$ of $R/I$
satisfies $J(n)=0$, then for every ideal $L$ of $R$ is either $L+I=R$ or
$L(n+1)=0$.

Proof. Apply Lemma 3.1 to the canonical projection $\pi\colon R\to R/I$. If
$L$ is an ideal of $R$ and $\pi(L)\neq R/I$, then $\pi(L(n))=0$, i.e.,
$L(n)\subseteq I$. Therefore $L(n+1)\subseteq I(1)=0$.

######  Lemma 3.3.

Let $R$ be a ring and let $M_{1},\dots,M_{k},I$ be ideals of $R$ such that
$(I+\cap_{j=1}^{k}M_{j})/I$ is a nil ideal of $R/I$. If $I\neq R$, then there
exists $1\leq j\leq k$ such that $M_{j}+I\neq R$.

Proof. Assume $M_{j}+I=R$ for every $1\leq j\leq k$. Then

$R=(I+M_{1})\cdots(I+M_{k})=I+M_{1}\cdots M_{k}\,.$

In particular, $1=i+m$ for some $i\in I$ and $m\in\cap_{j=1}^{k}M_{j}$. By our
assumption, $m+I$ is nilpotent in $R/I$ and hence $I=R$.

If $S$ is a ring and $\Delta$ is a set of derivations on $S$, then a set
$X\subseteq S$ is called $\Delta$-invariant if $\partial(X)\subseteq X$ for
every $\partial\in\Delta$. If $\Sigma$ is a set of ideals of $S$, maximal
elements of $\Sigma$ denotes the set of maximal elements of a partially
ordered set $(\Sigma\setminus\\{S\\},\subseteq)$.

######  Lemma 3.4.

Let $R$ be a ring, $\delta\colon R\to R$ a derivation on $R$, and
$S:=R[x,\delta]$ the corresponding ring of differential polynomials. Let
$\Delta$ be a set of derivations on $S$ satisfying the following conditions:

  1. (a)

$R$ is $\Delta$-invariant,

  2. (b)

$\partial(x)$ is a polynomial of degree at most $1$ for every
$\partial\in\Delta$,

  3. (c)

if $J\subseteq R$ is a $\Delta$-invariant ideal of $R$ such that
$\delta(J)\subseteq J$, then either $J=0$ or $J=R$.

Let $I$ be a nonzero proper $\Delta$-invariant ideal of $S$. Then

  1. (i)

There exists unique $f\in Z(S)$ monic such that $I=fS=Sf$.

  2. (ii)

If $R$ is right (left) artinian, then $S/I$ is right (left) artinian.

  3. (iii)

Let $\Sigma$ be the set of all $\Delta$-invariant ideals of $S$ containing
$I$. Then the set of maximal elements of $\Sigma$ is finite and if
$M_{1},\dots,M_{k}$ is a list of these elements, then $I=M_{1}^{e_{1}}\cdots
M_{k}^{e_{k}}$ for some $e_{1},\dots,e_{k}\in\mathbb{N}$.

Proof. Let $I$ be a nonzero proper $\Delta$-invariant ideal of $S$ and let
$d:={\rm min}\\{{\rm deg}(f)\mid 0\neq f\in I\\}$. Since $I\cap R$ is
$\Delta$-invariant by (a) and $\delta(I\cap R)\subseteq I\cap R$, (c) implies
$d>0$.

(i) (cf. [3, Proposition 2.1]) Consider the set $J:=\\{r\in R\mid\exists
r_{0},\dots,r_{d-1}\in R$ such that $rx^{d}+\sum_{i=0}^{d-1}r_{i}x^{i}\in
I\\}$. Then $J$ is easily checked to be an ideal of $R$. If
$f=\sum_{i=0}^{d}r_{i}x^{i}\in I$ and $\partial\in\Delta$, then, regarding
(b), $\partial(f)=\partial(r_{d})x^{d}+r_{d}\partial(x^{d})+g$, for some $g\in
S$ of degree at most $d-1$. So the coefficient by $x^{d}$ in $\partial(f)$ is
of the form $\partial(r_{d})+r_{d}s$ for some $s\in R$. Since $\partial(f)\in
I$ and $r_{d}\in J$, $\partial(r_{d})\in J$. That is, $J$ is invariant under
$\partial$. Also note that $xf-fx=\sum_{i=0}^{d}\delta(r_{i})x^{i}\in I$, thus
$\delta(r_{d})\in J$. By (c), $J=R$.

Therefore $I$ contains a monic polynomial $f$ of degree $d$. Moreover, it is
easy to see that $I$ is generated by $f$ as a left ideal. Observe that
polynomial $xf-fx$ has degree less than $d$ and is contained in $I$. It
follows that $xf-fx=0$, i.e, $f$ commutes with $x$. Similar argument shows
that $f$ commutes with every $r\in R$, hence $f\in Z(S)$. Since $I$ contains
no nonzero polynomials of degree less than $d$, $f$ is the only monic
polynomial of degree $d$ contained in $I$.

(ii) Assume that $R$ is left (right) artinian. Note that
$x^{i}+I,i=0,\dots,d-1$ generate $S/I$ as a left (right) $R$-module, so $S/I$
is left (right) artinian.

(iii) Let $\Sigma_{m}$ be the set of all maximal elements of $\Sigma$. If
$M\in\Sigma_{m}$, then, by (i), there exists a unique monic $f_{M}\in M$ such
that $M=f_{M}S$. Note that if $M\varsubsetneq N\in\Sigma$, then ${\rm
deg}(f_{M})>{\rm deg}(f_{N})$. In particular, $\Sigma$ satisfies a.c.c. We
claim that $f:=f_{I}$ is a product of elements from the set $\\{f_{M}\mid
M\in\Sigma_{m}\\}$.

Consider an equality $f=f_{M_{1}}\cdots f_{M_{k}}g$ with
$M_{1},\dots,M_{k}\in\Sigma_{m}$ and $g\in S$ of positive degree. Observe that
$f_{M_{1}},\dots,f_{M_{k}},g$ are monic and $f,f_{M_{1}},\dots,f_{M_{k}}\in
Z(S)$. Since monic polynomials are regular elements of $S$, $g\in Z(S)$. Let
$\partial\in\Delta.$ Since $I,M_{1},M_{2},\dots,M_{k}$ are $\Delta$-invariant,
there are $u,v_{1},v_{2},\dots,v_{k}\in S$ such that $\partial(f)=fu$,
$\partial(f_{M_{i}})=f_{M_{i}}v_{i}$ for $i=1,\dots,k$. Apply $\partial$ on
$f=f_{M_{1}}\cdots f_{M_{k}}g$:

$fu=\partial(f)=\sum_{i=1}^{k}f_{M_{1}}\cdots
f_{M_{i-1}}(f_{M_{i}}v_{i})f_{M_{i+1}}\cdots f_{M_{k}}g+f_{M_{1}}\cdots
f_{M_{k}}\partial(g)$ $fu=fh+f_{M_{1}}\cdots f_{M_{k}}\partial(g),$

where $h=\sum_{i=1}^{k}v_{i}$. Cancellation of the regular element
$f_{M_{1}}\cdots f_{M_{k}}$ from the equality above gives
$\partial(g)=g(u-h)$. In particular, the ideal $gS$ is invariant under
$\partial$ and hence $gS\in\Sigma$. Then there exists $M_{k+1}\in\Sigma_{m}$
containing $g$ and $g=f_{M_{k+1}}g^{\prime}$ for some $g^{\prime}\in S$ with
degree less than ${\rm deg}(g)$. After finitely many repetitions of this
process we get $f$ as a product of elements from the set $\\{f_{M}\mid
M\in\Sigma_{m}\\}$.

Let us change the notation and write $f=f_{M_{1}}^{e_{1}}\cdots
f_{M_{k}}^{e_{k}}$, where $M_{1},\dots,M_{k}$ are pairwise different elements
of $\Sigma_{m}$ and $e_{1},\dots,e_{k}$ are positive integers. Then
$fS=M_{1}^{e_{1}}\cdots M_{k}^{e_{k}}$. Let us verify that
$\\{M_{1},\dots,M_{k}\\}=\Sigma_{m}$. Assume that there exists $M\in\Sigma$
which is not contained in any element from $\\{M_{1},\dots,M_{k}\\}$. Since
$M+M_{i}$ is $\Delta$-invariant for every $1\leq i\leq k$, the maximality of
$M_{i}$ implies $M+M_{i}=S$. On the other hand, if
$M^{\prime}:=\cap_{i=1}^{k}M_{i}$, then $M^{\prime e}\subseteq fS$ for any
$e\geq\sum_{i=1}^{k}e_{i}$. This implies that $M^{\prime}+M/M$ is a nil ideal
of $S/M$ and therefore $S=M$ by Lemma 3.3. This shows that
$\Sigma_{m}=\\{M_{1},\dots,M_{k}\\}\,.$

######  Theorem 3.5.

Let $R$ be a noetherian prime ring such that the abelian group $(R,+)$ is
torsion free. Consider a ring of iterated differential polynomials
$S=R[x_{1},\delta_{1},x_{2},\delta_{2},\dots,x_{n},\delta_{n}]$ which
satisfies:

  1. (a)

$\delta_{i}(R)\subseteq R$ for every $i=1,\dots,n$.

  2. (b)

$\delta_{i}(x_{j})\in R[x_{1},\dots,x_{j}]$ for every $1\leq j<i\leq n$.

  3. (c)

If $J$ is an ideal of $R$ such that $\delta_{i}(J)\subseteq J$ for every
$i=1,\dots,n$, then $J=0$ or $J=R$.

Then $I(n)=0$ for every proper ideal $I$ of $S$.

Proof. We claim that it is enough to consider the case when $R$ is a simple
artinian ring of characteristic zero (and, in particular, a
$\mathbb{Q}$-algebra). Indeed, let $X$ be the set of all regular elements of
$R$. Then, by [5, Theorem 2.3.6], $X$ is a right denominator set in $R$.
Following the notation from Section 2, $X$ is also a right denominator set in
$S$ and the map $\varphi\colon S\to
RX^{-1}[x_{1},\partial_{1},\dots,x_{n},\partial_{n}]=SX^{-1}$ given by

$\varphi(r)=r.1^{-1},r\in R,\varphi(x_{i})=x_{i},i=1,\dots,n$

is a right localization of $S$ with respect to $X$. By Goldie’s theorem [3,
Theorem 6.18], $RX^{-1}$ is simple artinian and by Hilbert’s basis theorem [3,
Theorem 2.6], $SX^{-1}$ is left and right noetherian. Therefore it is possible
to apply [3, Theorem 10.18(a)] to see that $\varphi(K)X^{-1}$ is an ideal of
$SX^{-1}$ for every ideal $K$ of $S$. Also note that $(R,+)$ is a torsion free
abelian group, so $SX^{-1}$ is a $\mathbb{Q}$-algebra. Assume that every
proper ideal $U$ of $SX^{-1}$ satisfies $U(n)=0$. Consider a proper ideal
$I\subseteq S$. By (c), $I\cap R=0$, in particular $I\cap X=\emptyset$. Then
$\varphi(I)X^{-1}$ is a proper ideal of $SX^{-1}$ and, by Lemma 3.1, $I(n)=0$.

In the rest of the proof we assume that $R$ is a simple artinian
$\mathbb{Q}$-algebra. We proceed by induction on $n$. Let $n=1$ and consider a
nonzero proper ideal $I\subseteq S=R[x_{1},\delta_{1}]$. By Lemma 3.4(i)
applied for $\Delta=\emptyset$, $I$ is generated by a monic polynomial from
$Z(S)$. Then $I(1)=0$.

Now assume that the statement is true for rings with $n-1$ variables and
consider a proper ideal $I\subseteq
S=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$. Let $J:=I\cap R[x_{1}]$. We
distinguish two cases:

1) $J=0$. By [5, Theorem 1.2.9], $R[x_{1}]=R[x_{1},\delta_{1}]$ is a
noetherian prime ring, hence if $X$ the set of all its regular elements
$R[x_{1}]X^{-1}$ is a simple artinian ring. As in the first step of the proof,
$X$ is a right denominator set of $S$ and the homomorphism $\varphi\colon
S\to(R[x_{1}]X^{-1})[x_{2},\partial_{2},\dots,x_{n},\partial_{n}]$ given by

$\varphi(p)=p.1^{-1},p\in R[x_{1}],\varphi(x_{i})=x_{i},i=2,\dots,n$

is a right localization of $S$ with respect to $X$. Since $I\cap X=\emptyset$,
$\varphi(I)X^{-1}$ is a proper ideal of $SX^{-1}$ and
$\varphi(I)X^{-1}(n-1)=0$ by the induction hypothesis. Then $I(n-1)=0$ by
Lemma 3.1.

2) $0\neq J$. Observe that $J$ is a proper ideal of $R[x_{1}]$. By (a) and
(b), the restriction of $\delta_{i}$ to $R[x_{1}]$ is a ($\mathbb{Q}$-linear)
derivation of $R[x_{1}]$ and $\delta_{i}(J)\subseteq J$ for every $2\leq i\leq
n$. Let $P_{1},\dots,P_{k}$ be a list of all primes of $R[x_{1}]$ which are
minimal over $J$. By [5, Proposition 14.2.3], all these primes are stable
under $\delta_{2},\dots,\delta_{n}$. As explained in Section 2, for each
$i=1,\dots,k$ the ring $S/P_{i}S$ has a structure of a ring of iterated
differential polynomials,
$S/P_{i}S\simeq(R[x_{1}]/P_{i})[x_{2},\overline{\delta_{2}},\dots,x_{n},\overline{\delta_{n}}]$.
Also note that $R[x_{1}]/J$ is artinian by Lemma 3.4(ii) and $R[x_{1}]/P_{i}$
is a simple artinian ring for every $i=1,\dots,k$. When $S/P_{i}S$ is
considered as a ring of iterated differential polynomials, the appropriate
versions of conditions (a), (b), (c) for $S/P_{i}S$ are satisfied and we may
apply the induction. Therefore $L(n-1)=0$ for every proper ideal $L$ of
$S/P_{i}S$.

Repeating the argument from the first induction step, $P_{i}(1)=0$ and hence
also $P_{i}S(1)=0$ for every $i=1,\dots,k$. By Lemma 3.2, either $I(n)=0$ or
$I+P_{i}S=S$ for every $i=1,\dots,k$. Assume the latter and let
$P^{\prime}:=\cap_{i=1}^{k}P_{i}$. Since $(P^{\prime}+J)/J$ is the nilradical
of $R[x_{1}]/J$, there exists $m\in\mathbb{N}$ such that $P^{\prime
m}\subseteq J$. Also note $P^{\prime}S=\cap_{i=1}^{k}P_{i}S$ and
$(P^{\prime}S)^{m}=P^{\prime m}S\subseteq I$. Then $P^{\prime}S+I/I$ is a nil
ideal of $S/I$ and since we assume $P_{i}S+I=S$ for each $i=1,\dots,k$, Lemma
3.3 implies $I=S$. This contradicts that $I$ is a proper ideal of $S$ and
hence $P_{i}S+I\neq S$ for some $i$. Therefore $I(n)=0$.

######  Theorem 3.6.

Let $R$ be a right noetherian ring. Consider a ring of iterated differential
polynomials $S=R[x_{1},\delta_{1},x_{2},\delta_{2},\dots,x_{n},\delta_{n}]$
which satisfies

  1. (a)

$\delta_{i}(R)\subseteq R$ for every $i=1,\dots,n$,

  2. (b)

$1\leq j<i\leq n$ there exist $u,v\in R[x_{1},\dots,x_{j-1}]$ such that
$\delta_{i}(x_{j})=ux_{j}+v$,

  3. (c)

if $J$ is an ideal of $R$ such that $\delta_{i}(J)\subseteq J$ for every
$i=1,\dots,n$, then either $J=0$ or $J=R$.

Then $I(n)=0$ for every proper ideal $I$ of $S$.

Proof. Consider the ring $S_{0}:=R[x_{1}]\subseteq S$ and the set
$\Delta:=\\{\delta_{j}|_{S_{0}}\mid 2\leq j\leq n\\}$ of derivations on
$S_{0}$ induced by $\delta_{2},\dots,\delta_{n}$. Note that $0$ and $R$ are
the only $\delta_{1}$-invariant ideals of $R$ which are also
$\Delta$-invariant subsets of $S_{0}$ and $\partial(x_{1})$ is a polynomial of
degree at most one for every $\partial\in\Delta$. Hence Lemma 3.4 can be
applied to $S_{0}$ and $\Delta$.

We proceed by induction on $n$ along the lines of the proof of Theorem 3.5: If
$n=1$ and $I$ is a proper nonzero ideal of $S$, then $I$ is generated by a
monic polynomial of $Z(S)$ of positive degree, hence $I(1)=0$.

Now assume that the statement is true for rings with $n-1$ variables and
consider a proper ideal $I\subseteq
S=R[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$. Let $J:=I\cap S_{0}$. We
distinguish two cases:

1) $J=0$. Let $\Sigma$ be the set of all non-zero $\Delta$-invariant ideals of
$S_{0}$. For each $M\in\Sigma$ let $f_{M}\in Z(S_{0})\cap M$ be the monic
polynomial such that $f_{M}S_{0}=M$ (cf. Lemma 3.4(i)). Set $X:=\\{f_{M}\mid
M\in\Sigma\\}$. Since a product of $\Delta$-invariant ideals of $S_{0}$ is
$\Delta$-invariant, $X$ is a multiplicative set of regular elements of $S_{0}$
(observe $1=f_{S_{0}}$) and since $X\subseteq Z(S_{0})$, $X$ is a right
denominator set of $S_{0}$. Then $X$ is a right denominator set of regular
elements of $S$ and the canonical map $\varphi\colon S\to
S_{0}X^{-1}[x_{2},\partial_{2},\dots,x_{n},\partial_{n}]=SX^{-1}$ given by

$\varphi(s)=s.1^{-1},s\in S_{0},\ \varphi(x_{i})=x_{i},i=2,\dots,n$

is a right localization of $S$ with respect to $X$. Let us check, that the
inductive assumption can be applied to the ring of iterated differential
polynomials $S_{0}X^{-1}[x_{2},\partial_{2},\dots,x_{n},\partial_{n}]$, i.e.,
appropriate versions of (a),(b) and (c) are satisfied for
$S_{0}X^{-1}[x_{2},\partial_{2},\dots,x_{n},\partial_{n}]$. Conditions (a) and
(b) follow from the definition of derivations
$\partial_{2},\dots,\partial_{n}$. Let $L$ be a nonzero ideal of $S_{0}X^{-1}$
invariant under $\partial_{2},\dots,\partial_{n}$. Let $L_{0}=S_{0}\cap L$.
For $\ell\in L_{0}$ and $2\leq i\leq n$ is
$(\delta_{i}(\ell)).1^{-1}=\partial_{i}(\ell.1^{-1})\in L$. In particular,
$\delta_{i}(\ell)\in L_{0}$, i. e., $L_{0}\in\Sigma$. Then $f_{L_{0}}\in X$
and hence $L=L_{0}X^{-1}=S_{0}$.

Since $SX^{-1}$ is right noetherian, [3, Theorem 10.18(a)] implies
$\varphi(I)X^{-1}$ is an ideal of $SX^{-1}$. The assumption $J=0$ gives $X\cap
I=\emptyset$, i.e., $\varphi(I)X^{-1}$ is a proper ideal of $SX^{-1}$. The
inductive assumption applied to $SX^{-1}$ and Lemma 3.1 give that
$\varphi(I(n-1))=0$. Observe that $X$ consists of regular elements of $S$,
hence $\varphi$ is a monomorphism and therefore $I(n-1)=0$.

2) $J\neq 0$. By Lemma 3.4(i) $J=fS_{0}$ for some monic polynomial from
$Z(S_{0})$. Consider the set $\Sigma$ of all $\Delta$-invariant ideals of
$S_{0}$ which contain $J$ and let $\Sigma_{m}$ be the set of maximal elements
from $\Sigma$. If $M\in\Sigma$, then $MS=M[x_{2},x_{3},\dots,x_{n}]$ is an
ideal of $S$ and as noted in Section 2, $S/MS$ has a structure of a ring of
iterated differential polynomials,

$S/MS\simeq(S_{0}/M)[x_{2},\overline{\delta_{2}},\dots,x_{n},\overline{\delta_{n}}]\,.$

It follows directly from the construction in Section 2 that
$(S_{0}/M)[x_{2},\overline{\delta_{2}},\dots,x_{n},\overline{\delta_{n}}]$
satisfies appropriate versions of (a) and (b). Since $M$ is a maximal element
of $\Sigma$, is satisfies (c) as well. By induction, $L(n-1)$ for every proper
ideal $L$ of $S/MS$. By Lemma 3.4(iii), the set $\Sigma_{m}$ is finite and by
Lemma 3.4(i), $M(1)=0$ for every $M\in\Sigma_{m}$. Therefore $MS(1)=0$ for
each $M\in\Sigma_{m}$. Moreover, if $\Sigma_{m}=\\{M_{1},\dots,M_{k}\\}$, then
$J=M_{1}^{e_{1}}\cdots M_{k}^{e_{k}}$ for some
$e_{1},\dots,e_{k}\in\mathbb{N}$ and $(\cap\Sigma_{m})^{e}\subseteq J$ for
every $e\geq\sum_{i=1}^{k}e_{i}$. By Lemma 3.2, either $I(n)=0$ or $I+MS=S$
for every $M\in\Sigma_{m}$. Assume the latter and let
$N:=\cap_{i=1}^{k}M_{i}S$. Then $N+I/I$ is a nil ideal of $S/I$ and, by Lemma
3.3, $I=S$. This contradiction proves that $I(n)=0$.

######  Remark 3.7.

Note that if $S$ is as in Theorem 3.6, then every nonzero ideal of $S$
contains a regular element. Indeed, keeping the notation from the proof of
Theorem 3.6, we use the same inductive arguments: If $n=1$, then $I$ contains
a monic polynomial. If $n\geq 2$ and $J\neq 0$, then $J$ contains a monic
polynomial. If $n\geq 2$ and $J=0$, then, by induction, $\varphi(I)X^{-1}$
contains a regular element $\varphi(f)\varphi(g)^{-1}$, where $f\in I$ and
$g\in X$. But since $\varphi$ is a monomorphism and $\varphi(g)^{-1}$ is
invertible in $SX^{-1}$, $f$ is a regular element of $S$.

Let $L$ be a Lie algebra of finite dimension over a field $\mathbb{F}$. Recall
that $L$ is solvable if it has a chain of subalgebras $0=L_{0}\subset
L_{1}\subset\cdots\subset L_{n}=L$ such that ${\rm dim}\ L_{i}=i$ and $L_{i}$
is an ideal in $L_{i+1}$ for each $i=0,\dots,n-1$. If there exists such a
chain consisting of ideals in $L$, then we say that $L$ is completely
solvable. If subalgebras in this chain satisfy $[L,L_{i}]\subseteq L_{i-1}$
for $i=1,2,\dots,n$, then $L$ is said to be nilpotent.

If $L$ is completely solvable, then its universal enveloping algebra $U(L)$
has a structure of a ring of iterated differential polynomials
$U(L)=\mathbb{F}[x_{1},\delta_{1},\dots,x_{n},\delta_{n}]$ (cf. [3, p. xx])
which satisfies assumptions of Theorem 3.6. Combining Theorem 3.6 and
Proposition 1.1, we get

######  Corollary 3.8.

Let $L$ be a completely solvable Lie algebra of dimension $n$ over a field
$\mathbb{F}$. Then $I(n)=0$ for any proper ideal $I$ of $U(L)$. In particular,
every projective $U(L)$-module is either finitely generated or free.

Recall that, by [6, Proposition 4.1], $I(2)=0$ if $I$ is a proper ideal of
$U(L)$, where $L$ is a finite-dimensional solvable Lie algebra over a field of
characteristic $0$. In the last part of this paper we show a similar result
for ideals of finite codimension in $U(L)$, where $L$ is a finite-dimensional
Lie algebra over an arbitrary field with $[L,L]$ nilpotent. If the
characteristic of the field is zero, then $[L,L]$ is nilpotent for any
solvable Lie algebra of finite dimension by [5, Theorem 14.5.3]. The argument
of [5, Theorem 14.5.3] also shows $[L,L]$ is nilpotent if $L$ is completely
solvable and the converse is true if the field is algebraically closed of
characteristic zero.

######  Lemma 3.9.

Let $R$ be a right artinian ring, $\Delta$ a set of derivations on $R$ and let
$\Sigma$ be the set of all $\Delta$-invariant ideals of $R$. Then the set
$\Sigma_{m}$ of all maximal elements of $\Sigma$ is finite and
$\cap\Sigma_{m}$ is nilpotent.

Proof. Let $k$ be the length of $R_{R}$. If $M_{1},\dots,M_{\ell}$ are
pairwise different elements of $\Sigma_{m}$, then the Chinese reminder theorem
implies that $\prod_{i=1}^{\ell}R/M_{i}$ is a factor of $R$, hence $\ell\leq
k$. Therefore $|\Sigma_{m}|\leq k$.

Let $J(R)$ be the Jacobson radical of $R$ and let $\pi\colon R\to R/J(R)$ be
the canonical projection. We claim that for every maximal ideal $I$ of
$R/J(R)$ there exists $M\in\Sigma_{m}$ such that $\pi(M)=I$. Indeed, let $I$
be a maximal ideal of a semisimple artinian ring $R/J(R)$. Then there exists
an idempotent $e^{\prime}\in R/J(R)$ such that $I=e^{\prime}R/J(R)$. Since
idempotents can be lifted modulo $J(R)$ (cf. [1, Proposition 27.1]), there
exists an idempotent $e\in R$ such that $e^{\prime}=\pi(e)$. Then
$M^{\prime}:=ReR$ is $\Delta$-invariant and hence it is contained in some
$M\in\Sigma_{m}$. Obviously $\pi(M^{\prime})$ contains $I$, so either
$\pi(M)=I$ or $\pi(M)=R/J(R)$. The latter would mean $M+J(R)=R$ and
consequently $M=R$ which is not the case. Therefore $\pi(M)=I$.

Finally let $J:=\cap\Sigma_{m}$. Then, by the claim, $\pi(J)$ has to be
contained in every maximal ideal of $R/J(R)$. In other words, $\pi(J)\subseteq
J(R)$ which is a nilpotent ideal since $R$ is right artinian.

Recall that an ideal $I$ of a right noetherian ring $R$ has the right AR-
property if for every right ideal $A$ of $R$ there exists $m\in\mathbb{N}$
such that $A\cap I^{m}\subseteq AI$. A sequence $r_{1},r_{2},\dots,r_{n}\in R$
is said to be a centralizing sequence if for each $j\in\\{1,\dots,n\\}$ the
image of $r_{j}$ in $R/\sum_{i=1}^{j-1}r_{i}R$ is a central element and
$R\neq\sum_{i=1}^{n}r_{i}R$. By [5, Theorem 4.2.7(ii)], any ideal of a right
noetherian ring with a centralizing sequence of generators has the right AR-
property.

######  Proposition 3.10.

Let $R$ be a right artinian ring. Consider a ring of iterated differential
polynomials $S=R[x_{1},\delta_{1},x_{2},\delta_{2},\dots,x_{n},\delta_{n}]$
which satisfies

  1. (a)

$\delta_{i}(R)\subseteq R$ for every $i=1,\dots,n$,

  2. (b)

$\delta_{i}(x_{j})\in R$ for every $1\leq j<i\leq n$,

  3. (c)

if $J$ is an ideal of $R$ such that $\delta_{i}(J)\subseteq J$ for every
$i=1,\dots,n$, then either $J=0$ or $J=R$.

If $I$ is a proper ideal of $S$ such that the right $R$-module $S/I$ has
finite length, then $I$ is contained in a proper ideal of $S$ having the right
AR-property. Moreover, $I(1)=0$.

Proof. Let $I$ be a proper ideal of $S$ such that $S/I$ is a right $R$-module
of finite length. We claim that $I$ is contained in a proper ideal $K$ of $S$
which satisfies the right AR-property. More precisely, by induction on $n$ we
show that $I$ is contained in an ideal of $S$ which can be generated by a
centralizing sequence.

If $n=1$, then $I\neq 0$ and $I$ is generated by a monic polynomial $f\in
Z(S)$ by Lemma 3.4(i). In this case we just set $K:=I$.

Assume $n>1$ and let $S_{0}:=R[x_{1}]$, $J:=I\cap S_{0}$. Note that $S_{0}/J$
is an $R$-submodule of $S/I$, hence $S_{0}/J$ is a right artinan ring. In
particular, $J\neq 0$. Let $\Delta:=\\{\delta_{i}|_{S_{0}}\mid 2\leq i\leq
n\\}$ be the set of derivations on $S_{0}$ induced by derivations
$\delta_{2},\dots,\delta_{n}$. Further let $\Sigma$ be the set of all
$\Delta$-invariant ideals of $S_{0}$ which contain $J$ and let $\Sigma_{m}$ be
the set of all maximal elements of $\Sigma$. By Lemma 3.4(i), each
$M\in\Sigma$ is generated by a unique monic polynomial $f_{M}\in Z(S_{0})$
which is a non-zero polynomial of $M$ of the smallest possible degree. But
regarding (b), ${\rm deg}(\delta_{i}(f_{M}))<{\rm deg}(f_{M})$ for every
$2\leq i\leq n$. Hence $\delta_{i}(f_{M})=0$. Overall $f_{M}$ commutes not
only with elements of $S_{0}$ but also with $x_{2},\dots,x_{n}$. In other
words, $f_{M}\in Z(S)$.

By Lemma 3.9 applied to $S_{0}/J$ and the set of derivations
$\overline{\Delta}$ on $S_{0}/J$ which are induced by derivations of $\Delta$,
the set $\Sigma_{m}$ is finite. Let $\Sigma_{m}=\\{M_{1},\dots,M_{k}\\}$ and
$L:=\cap_{i=1}^{k}M_{i}$. Then there exists $\ell\in\mathbb{N}$ such that
$L^{\ell}\subseteq J$. For $i=1,\dots,k$ let $M_{i}^{\prime}$ be the ideal of
$S$ generated by $M_{i}$. Then $L^{\prime}:=\cap_{i=1}^{k}M_{i}^{\prime}=LS$
and $L^{\prime\ell}=L^{\ell}S\subseteq I$. In particular, $L^{\prime}+I/I$ is
a nil ideal of $S/I\neq 0$ and by Lemma 3.3 there exists $i\in\\{1,\dots,k\\}$
such that $M_{i}^{\prime}+I\neq S$.

Let $M\in\Sigma_{m}$ be such that $MS+I\neq S$. Recall that $M=f_{M}S_{0}$ for
some $f_{M}$ in $Z(S)$. Using the notation introduced in Section 2, let

$\pi\colon
S\to(S_{0}/M)[x_{2},\overline{\delta_{2}},\dots,x_{n},\overline{\delta_{n}}]=:\overline{S}$

be the homomorphism sending the term $rx_{1}^{e_{1}}x_{2}^{e_{2}}\cdots
x_{n}^{e_{n}}$ to $(rx_{1}^{e_{1}}+M)x_{2}^{e_{2}}\cdots x_{n}^{e_{n}}$. Note
that ${\rm Ker}\ \pi=MS$ and $\pi(I)$ is a proper ideal of $\overline{S}$ and
$\overline{S}/\pi(I)$ is finitely generated right $S_{0}/M$-module. Also the
appropriate versions of (a), (b) and (c) are valid for $\overline{S}$: (a) and
(b) are easy to check and (c) is satisfied since $M\in\Sigma_{m}$. By the
induction, $\pi(I)$ is contained in $K^{\prime}$ a proper ideal of
$\overline{S}$ which is generated by a centralizing sequence
$\pi(s_{2}),\dots,\pi(s_{t})$ for some $s_{2},\dots,s_{t}\in S$. Let
$K:=\pi^{-1}(K^{\prime})$, $s_{1}:=f_{M}$. Then $s_{1},s_{2},\dots,s_{t}$ is a
centralizing sequence of elements in $S$ generating $K$. Since $I\subseteq K$,
we are done.

By [7, Theorem 2.2], $\cap_{i\in\mathbb{N}}K^{i}=\\{s\in S\mid\exists a\in K$
such that $0=s(1-a)\\}$. If $K(1)=\cap_{i\in\mathbb{N}}K^{i}\neq 0$, then
$K(1)$ contains a regular element $r$ by Remark 3.7. But then $r(1-a)=0$ for
$a\in K$ implies $K=S$, which is not the case. Therefore $K(1)=0$, hence also
$I(1)=0$.

######  Proposition 3.11.

Let $L$ be a finite dimensional Lie algebra over any field such that $[L,L]$
is nilpotent. If $I$ is a proper ideal of finite codimension in the universal
enveloping algebra $U(L)$, then $I(2)=0$.

Proof. Let $N:=[L,L]$ be the commutator of $L$ and let $R:=U(N)$ be the
universal enveloping algebra of $N$. Let
$x_{1},\dots,x_{k},x_{k+1},\dots,x_{n}$ be a basis of $L$ such that
$x_{1},\dots,x_{k}$ is a basis of $N$, then
$S:=U(L)=R[x_{k+1},\delta_{k+1},\dots,x_{n},\delta_{n}]$ where
$\delta_{i}(x_{j})=[x_{i},x_{j}]\in N\subseteq R$ for every $k\leq j<i\leq n$.
Let $I$ be a proper ideal of $U(L)$ such that $U(L)/I$ is a finite dimensional
algebra. Let $J:=R\cap I$. Observe that $J$ is an ideal of $R$ of finite
codimension invariant under $\delta_{k+1},\dots,\delta_{n}$. In particular,
$R/J$ is an artinian ring. Let $\Delta=\\{\delta_{i}|_{R}\mid
i=k+1,\dots,n\\}$ be the set of derivations on $R$ given by restrictions of
$\delta_{k+1},\dots,\delta_{n}$. Consider the set $\Sigma$ of all
$\Delta$-invariant ideals of $R$ which contain the ideal $J$. By Lemma 3.9
applied to $R/J$ and the set of derivations on $R/J$ induced by derivations
from $\Delta$, we see that the set $\Sigma_{m}$ of maximal elements of
$\Sigma$ is finite and there exists $m\in\mathbb{N}$ such that
$(\cap\Sigma_{m})^{m}\subseteq J$. Repeating the arguments from the proof of
Proposition 3.10, there exists $M\in\Sigma_{m}$ such that $MS+I\neq S$: Let
$\Sigma_{m}=\\{M_{1},\dots,M_{\ell}\\}$, $T:=\cap\Sigma_{m}$. Then
$\cap_{i=1}^{\ell}M_{i}S=TS$ and $(TS)^{m}\subseteq JS\subseteq I$, thus
$I+TS/I$ is a nil ideal of $S/I$. By Lemma 3.3, there exists $M\in\Sigma_{m}$
such that $MS+I\neq S$.

Let $M\in\Sigma_{m}$ be such that $MS+I\neq S$. Using the notation of Section
2, the ring $\overline{S}:=S/MS$ can be seen as a ring of differential
polynomials
$\overline{S}=R/M[x_{k+1},\overline{\delta_{k+1}},\dots,x_{n},\overline{\delta_{n}}]$.
Denote $\overline{I}:=I+MS/MS$ the image of $I$ in $S/MS$. Let
$\overline{\Delta}=\\{\overline{\delta_{i}}|_{R/M}\mid i=k+1,\dots,n\\}$ be
the set of derivations on $R/M$ given by the restrictions of
$\overline{\delta_{k+1}},\dots,\overline{\delta_{n}}$. Note that the only
$\overline{\Delta}$-invariant ideals of $R/M$ are the trivial ones, therefore
we can apply Proposition 3.10 to $\overline{I}$. That is, $\overline{I}(1)=0$
and therefore $I(1)\subseteq MS$ by Lemma 3.1.

Finally, recall that the Lie algebra $N$ is nilpotent and therefore, by [4,
Theorem 4.2], $R=U(N)$ is a right AR-ring, i.e., every right ideal of $R$ has
the right AR-property. By [7, Theorem 2.2], $M(1)=0$ and also $MS(1)=0$. Then
$I(2)\subseteq MS(1)=0$.

######  Corollary 3.12.

Let $V$ be a simple finite-dimensional module over a finite-dimensional Lie
algebra with nilpotent commutator. Then ${\rm Ext}(V,V)\neq 0$.

Proof. Consider $V$ as a (right) module over $U(L)$. Its annihilator $I$ is an
ideal of $U(L)$ of finite codimension. If ${\rm Ext}(V,V)=0$, then arguments
from the proof of [6, Corollary 3.8] show that $I=I^{2}$ which contradicts
Proposition 3.11.

## References

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# A ‘simple metal’ description of liquid carbon, its warm-dense matter states,
and the identification of liquid-liquid phase transitions.

M. W. C. Dharma-wardana<EMAIL_ADDRESS>National
Research Council of Canada, Ottawa, Ont. Canada K1A 0R6  Dennis D. Klug
National Research Council of Canada, Ottawa, Ont. Canada K1A 0R6

###### Abstract

Liquid carbon is a complex fluid whose theory is claimed to require explicit
covalent interactions and many-center potentials. Thus very expensive $N$-atom
quantum simulations for $N\sim 100-500$ using density-functional theory (DFT),
and molecular-dynamics (MD) are usually deployed. They show intriguing
structure factors with a split first peak and features not found in simple
metallic liquids. We show that a simple-metal model using only one-body
electron densities, one-body ion densities, and appropriate exchange-
correlation functionals implemented in the one-atom DFT approach of the
neutral pseudo-atom (NPA) model, quantitatively and inexpensively recovers the
results of $N$-atom DFT simulations. We show that structural changes are
dominated by strong electron-ion interactions at the Fermi energy. Evidence is
presented for three liquid-liquid phase transitions in the 3 to 4 g/cm3 range
using NPA calculations supported by DFT-MD simulations.

###### pacs:

62.50.-p, 52.25.Fi, 81.05U-, 78.70.Ck

Introduction \- Many light elements such as H, B, C, N, Si, P, and their
mixtures form strong covalency in the solid. They seem to retain some of this
bonding on melting galli89 ; Stich89 ; glosli99 ; ghiring05  and even in warm-
dense-matter (WDM) regimes Vorberger20 ; whitley15 ; kraus13 ; hammelCH12 ;
DWP-Carb90 . A theory for reliably and inexpensively predicting the properties
of these materials is needed in technological applications. Their properties
under high densities $\bar{\rho}$ and temperatures $T$ are needed in high-
energy density physics lindl04 ; benedict2014  and astrophysics hubNellis91 ;
Potekhin05 ; sherman12 . Besides technological interests, the quantum theory
of its liquid state has provoked great interest since the 1980s galli89 ; DWP-
Carb90 .

We focus on basic issues of wider interest and the intriguing interactions in
“tetrahedral fluids”, typified by liquid carbon ($l$-C) where such effects are
strongest. A large effort, using chemical bonding models treat these electron-
ion systems via multi-center ion-ion potentials, with the electron subsystem
integrated out. Within such models, pair-potentials cannot stabilize diamond
structures without at least “three body forces”, as in, say, the Stillinger-
Weber model SW85 .

This paradigm uses potentials with hundreds of parameters fitted to data
covering wide ranging densities $\bar{\rho}$, temperatures $T$ and structures.
Yet, even a model for liquid aluminum at finite $T$, a trivial problem in
standard liquid-metal theory, requires over forty parameters in current
effective medium theories Ercolessi94 . Nevertheless, these multi-center
potentials give disappointing results for liquid carbon kraus13 . Glossli et
al. glosli99  using the Brenner potential found a liquid-liquid phase
transition (LPT) between a $sp^{3}$ bonded high density liquid and an
$sp,sp^{2}$ bonded low density liquid. However, a DFT-MD study using the
Perdew-Burke-Ernzerhof (PBE) exchange-correlation (xc) functional PBE96  found
no such LPT WuLPT02 ; glosli99 . Here we report LPTs revealed by one-center
DFT methods and confirmed by standard $N$-center DFT-MD simulations.

The DFT-MD method implemented in codes, e.g., VASP VASP  or ABINIT ABINIT
will be denoted ‘Quantum MD’ (QMD) for brevity. It treats the liquid as an
average over many realizations of a periodic cluster of $N$-atoms in the
simulation cell. It is an $N$-center DFT for an $N$-center electron density
$n(\vec{r},\vec{R}_{1},\cdots,\vec{R}_{N})$. Such complex densities are
sensitive to the electron xc-functional used. Remsing et al. Remsing17  showed
that the VASP results for $l$-Si from the PBE functional differed
significantly from those of the ‘strongly constrained and appropriately
normed’ (SCAN) functional SCAN13 . Expensive force-matching methods
Ercolessi94 ; Stanek21  and decompositions of $N$-body properties Li-Parr86 ;
BethkenZbar20  are needed to extract individual atomic or pair properties from
QMD calculations.

According to Hohenberg and Kohn, one-body densities of electrons and ions are
sufficient for a complete thermodynamics of any electron-ion system. A one-
atom DFT is utilized in the neutral-pseudo-atom (NPA) method used here. It is
an ‘average-atom’ model RozsnyaiSigma2008 ; Murillo13 , but not restricted to
high $T$. It was recently used for supercooled $l$-Si, a complex ‘tetrahedral’
liquid cdwSi20 ; eos95 ; Pe-Be . The many-ion and many-electron effects are
included in the NPA via suitable ion-ion, electron-electron and electron-ion
exchange-correlation (xc) functionals of the one-body electron density
$n(\vec{r})$ and the one-body ion density $\rho(\vec{r})$ DWP82 ; cdw-pop21 .

Unlike the Si4+ ion with a robust core, the point-like C4+ ion produces strong
interactions, especially in the low-$\bar{\rho}$, low-$T$ region. Stanek et
al. Stanek21  showed that the use of linear-response (LR) potentials generated
from NPA for $l$-C at the “graphite density” $\rho_{\rm G}\simeq$ 2.267g/cm3
at low $T$ strongly over-estimated the first peak in the NPA pair distribution
function (PDF). Linear respons is known to fail for expanded metals at low
$T$.

In the NPA a carbon nucleus is at the origin of $l$-carbon, a uniform system
with a free electron density $\bar{n}$. It acquires a bound electron density
$n_{b}(r)$ within its Wigner-Seitz sphere of radius $r_{\rm ws}$, a free
electron distribution $n_{f}(r)$ and an ion distribution $\rho(r)$ covering
the whole volume, with a mean density $\bar{\rho}$.

QMD and NPA calculations show that the average ionization $\bar{Z}=4$ for the
$\bar{\rho},T$ studied here Vorberger20 ; galli89 . The C-C bonds (single,
double etc.,) have a bond energy $E_{b}$ of the order of $\sim$ 4-8 eV Bond-E
. However, a WDM medium containing free electrons produces screening effects
causing the band gap to close on melting, with the conductivity and
reflectivity sharply increasing. However, a peak at the C-C distance of 1.4 Å
is found in the $g_{\rm cc}(r)$ QMD simulations galli89  and even in early NPA
calculations DWP-Carb90 .

The NPA calculation provides the free electron density $n_{f}(r)$ and the
excess free electron density $\Delta n_{f}(r)=n_{f}(r)-\bar{n}$ at a carbon
ion carrying its bound shell of electrons, as discussed in more detail in SMSM
which is given here as an Appendix. The Fourier transform $\Delta n_{f}(k)$
serves to construct pair potentials.

Pseudopotentials and pair potentials in $l$-C – The $N$-center QMD
calculations provide an $N$-center potential energy surface used to construct
force-matched pair-potentials. The electrons have been eliminated from such
potentials Ercolessi94 . In contrast, the NPA treats a two-component system of
electrons and ions, and forms a pseudopotential $U_{ei}(k)$ for the specified
$\bar{n},T$(see Fig. 1). The simplest local linear-response (LR) potential is:

$U_{ei}(k)=\frac{\Delta
n_{f}(k)}{\chi(k,\bar{n},T)},\;U_{ei}(k)=-ZV_{k}M_{k},$ (1)

Here $\chi(k)$ is the linear response of the interacting electron fluid at
$\bar{n},T$, discussed further in the Appendix given as Supplemental Material
SM . The Coulomb potential $V_{k}$ and the form factor are $4\pi/k^{2}$,
$M_{k}$ respectively. The pseudopotential is in LR, but $\Delta n_{f}(k)$ has
all the non-linear effects of the Kohn-Sham calculation. The nonlinearities in
the fluid response are not treated as the response function $\chi(k)$ is based
on plane waves.

Figure 1: (Color online) (a) The LR-NPA pseudopotential $U_{ei}(k)$ from Eq. 1
for $l$-carbon at 1 eV for 2.5g/cm${}^{3}\leq\bar{\rho}\leq$ 10g/cm3. The
$U_{ei}$ is invariant betwenn 5-10g/cm3 and a sharp change occurs below
5g/cm3. The $U_{ei}(k=2k_{F})\to 0$ at $\rho_{\rm D}\simeq 3.5$g/cm3 (line
with circles). Panel (b) shows typical $S(k)$ using the LR-$U_{ei}(k)$ and the
electron fluid response function $\chi_{ee}(k,T,\bar{n})$. The unconnected
symbols for $\bar{\rho}\geq 5$g/cm3 are the QMD $S(k)$ of Ref. Vorberger20
using the PBE functional. The smooth curves are NPA calculations. The high-$K$
subpeak (marked Hk) falls on 2$k_{F}$ for $\bar{\rho}\geq 3$g/cm3. We compare
the LR-NPA $S(k)$ and QMD-SCAN $S(k)$ at 3 g/cm3, and 2.5 g/cm3 showing that
LR-NPA is unacceptable below 3 g/cm3.

The limits of validity of the LR $U_{ei}(k)$ are discussed in Ref. Utah12 . It
fails for low $\bar{\rho},T$ when non-linear effects prevail.

In Fig. 1(a) we display the pseudopotential for a range of densities, for $T$
= 1 eV. With $Z=4$, $l$-C is a dense electronically degenerate metal even at 1
eV, even for 2.5g/cm3 when $r_{s}\simeq 1.48$ and $E_{F}\simeq 23$ eV.
Essentially all electron-ion scattering occurs at the Fermi energy $E_{F}$,
with a momentum transfer $q=2k_{F}$, where $K_{F}$ is the Fermi momentum.

A sharp change in $M_{k}$ occurs just below 5g/cm3 as seen from Fig. 1(a).
Then, near the ‘diamond density’ $\rho_{\rm D}$, viz., $\simeq$ 3.52g/cm3
$M_{k}$ at $k=2k_{F}$ passes through zero and changes sign (curve with
circles). This suggests that the LR $U_{ei}(k)$, Eq. 1, begins to fail for
$\bar{\rho}\leq\rho_{\rm D}$ even though the electron density is high.
However, we display the linear-response $U_{ei}(k)$ and the calculated $S(k)$
even for densities below $\rho_{\rm D}$ in Fig. 1, as they indicate when non-
linear effects become important in the range $\rho_{\rm
D}>\bar{\rho}\geq\rho_{\rm G}$. Figure 1 shows that LR is unacceptable below
$\bar{\rho}\sim 3$ g/cm3 and $T$ = 1 eV.

The LR pseudopotential defines a pair potential in second-order perturbation
theory:

$V_{ii}(k)=Z^{2}V_{k}+|U_{ei}(k)|^{2}\chi(k,r_{s},T)$ (2)

This implies that at $\rho_{\rm D}$ the pair potential at 2$k_{F}$ becomes
just $Z^{2}V_{k_{F}}$ since $M_{2k_{F}}=0$. The pair potential is that of two
unscreened carbon ions if non-linear effects are neglected. Our NPA and QMD
calculations detect an LPT very near this density.

The ion distribution $\rho(r)=\bar{\rho}g(r)$ is given by the ion-DFT equation
using the pair-potential $V_{ii}(r)$ and the ion-ion xc-functional. This
amounts to solving a hyper-netted-chain (HNC) equation containing a bridge
function DWP82 , to yield the $g(r)$ and the $S(k)$ of the ion subsystem. In
Fig. 1(b) we compare the NPA and QMD results of Ref. Vorberger20  where the
PBE xc-functional has been used.

The first peak of $S(k)$ displays lower-$k$ (Lk) and higher-$k$ (Hk) subpeaks.
The interactions occur via strong electron scattering at the Fermi surface,
with a momentum transfer of $q=2k_{F}$. The fluid responds by creating the
high-$k$ peak Hk as close to 2$k_{F}$ as possible for fluids with
$\bar{\rho}>\sim 3$ g/cm3 as shown in panel (b). This occurs if enough atoms
occupy the first turning point of the Friedel oscillations of the pair-
potential, while excess atoms move to the lower-$k$ (i.e., more distant)
weaker minima, and vice versa depending on the fluid density.

The electron xc-energy favours a high-density electron fluid, i.e., a shell of
atoms even in a positive energy ledge of the pair-potential (Fig. 2). This is
less advantageous at lower density and the peak Hk at $2k_{F}$ grows at the
expense of Lk, as seen in Fig. 1. The LR-pseudopotential at $2k_{F}$, i.e.,
$U_{ei}(2k_{F})$ becomes zero at about the diamond density, $\rho_{\rm D}\sim
3.5$g/cm$3$, and changes sign for lower densities. Coherent scattering from
atoms positioned in the Friedel minima contribute coherently to the subpeak Hk
in $S(k)$ at 2$k_{F}$. This coherence links the coordination number with the
liquid phases of a homogeneous fluid.

The ion distribution modifies itself to minimize the energy of the system in
two ways: (i) its Hk peak locates itself as closely as possible to 2$k_{F}$.
(ii) At densities below $\rho_{\rm D}$, the first atomic shell attempts to
still retain a high coordination number $N_{c}$ at the expense of outer
shells. Even when $\bar{\rho}$ is low, the local density of the fluid adjusts
to bring Hk to 2kF, lowering the free energy via strong scattering at $E_{F}$.
That is, atoms may be drawn towards the central ion by decreasing the depths
of secondary Friedel minima. This is favourable for $N_{c}\geq 6$.

Figure 2: Color online (a) A comparison of the C-C pair potentials from a
force-matched DFT-MD calculations at $\rho_{3}$=3.27g/cm3 whitley15  with the
NPA pair potential in LR. The force-matched DFT-MD potentials fail to recover
the higher Friedel oscillations (for $r>5$ a.u.) seen in the NPA potentials.
(b)The same data are shown on a log-scale (y-axis). The maximum of the $g(r)$
at this density corresponds to the positive energy ledge for this range of
densities. The extent of the agreement of the the $g(r),S(k)$ from the NPA and
the QMD simulation whitley15  for this case was discussed in Ref. CPP-carb18

Evidence for LPTs in liquid carbon. Here we limit ourselves to the
$\bar{\rho}\geq 2.9$ g/cm3 range where the LR forms for
$U_{ei}(k),V_{ii}(k),S(k)$ etc., adequately agree with QMD results without
non-LR modifications. We examine the pressure and the compressibility for
signs of LPTs using the NPA for $\bar{\rho}>2.9$g/cm3, thus excluding the low
$\bar{\rho}$ explored in Ref. vanThiel93 . We find three LPTs labeled LPT3,
LPT3.5, and LPT3.75, as they occur at $\bar{\rho}\simeq$ 3.15, 3.5, and 3.75
g/cm3 in the NPA calculations. The QMD results confirm them, at slightly
shifted $\bar{\rho}$ values. Thus the LPT3.5 occurs at 3.42g/cm3 in the QMD-
SCAN data.

Figure 3: (Color online) (a) The pressure at 1 eV from QMD-PBE Vorberger20 ,
QMD-SCAN and NPA calculations. The $k\to 0$ of $S(k)$, viz.,
$S(0)=\kappa/\kappa^{0}$ where $\kappa$ is the isothermal compressibility
displays discontinuities (arrows) at the $P$-discontinuities. The pressure at
10 eV shows no discontinuities. (b) The discontinuities in the NPA pressure at
1 eV support three LPTs at $\bar{\rho}\sim 3.15$, $\sim 3.52$, and 3.75 g/cm3.
The data points above and below $\bar{\rho}\simeq$ 3.42g/cm3 from QMD-SCAN
accurately fall on straight lines intersecting at $\bar{\rho}\simeq$ 3.42g/cm3
if a single Gaussian is fitted to the QMD $P$-distributions. However, the
spread of the SCAN-QMD $P$ distributions (labeled SCAN-$\Delta P$,
$\times$3.3) increase (arrows) at LPT3 and LPT3.75 and use a two-Gaussian fit
(see Fig. 4).

Figure 3(a) displays the pressure from the $N$-atom QMD calculation of Ref.
Vorberger20 , using $N$ = 196 to 256 atoms and the PBE xc-functional. Our QMD
calculation for $\bar{\rho}<4$g/cm3 using the SCAN functional with $N=108$
atoms is also shown. The NPA, PBE and SCAN calculations of $P$ differ by about
the same magnitude in this region. It also displays the NPA pressure and the
isothermal compressibility via the limit $S(k\to 0)$, i.e.,
$S(0)=\kappa/\kappa^{0}$. Here $\kappa^{0}=1/\bar{\rho}T$ is the ideal fluid
compressibility. The $S(0)$ calculation is independent of the pressure
calculation. Hence discontinuities in the pressure, and in $S(0)$
independently support one another. The $P$ data at 10 eV show no
discontinuities.

Fig 3(b) displays the $2.9\leq\bar{\rho}\leq 4$ g/cm3 range. The QMD-SCAN
pressure confirms the LPT3.5 seen in the NPA, while the other two LPTs seen in
the NPA require a deeper examination of the QMD-SCAN pressure data.

The fluctuations in the pressure within an equilibriated MD run fall on a
Gaussian distribution (see SMSM , i.e., the Appendix) for a uniform fluid. The
width of $P$ fluctuations in the QMD-SCAN (labeled SCAN-$\Delta P$) change
significantly near the discontinuities (vertical arrows, Fig. 3) at LPT3 and
LPT3.75. It is found that the $P$ distributions at LPT3 and LPT3.75 are best
fitted with two Gaussians instead of one, giving two pressure peaks (Fig.4).
At LPT3.5 the distribition is not bimodal (see SM, or Appendix). Thus at
3.1g/cm3, two contributions at 70% and 30% give the average QMD-SCAN $P$ shown
in Fig. 3. The need for two-Gaussians at LPT3 and LPT3.75 to describe the QMD
$P$ confirms the NPA pressure discontinuities for a uniform fluid. QMD
averages over clusters, does not demand microscopic uniformity, and fails to
reveal these LPTs explicitly.

Figure 4: The $P$-distribution from QMD-SCAN calculations. (a) 2.9g/cm3 and
(b) 3.1g/cm3, at LPT3. While the 2.9g/cm3 case fits to one Gaussian, that at
3.1g/cm3 is best fitted with two Gaussians, implying an inhomogeneous plasma
at the LPT. This is also the case for LPT3.75, but the LPT3.5 is not
considered bimodal (for more details, see Appendix).

There are corresponding discontinuities in $S(0)$ since the fluid
compressibility changes sharply at the LPTs, while $\bar{Z}$ remains constant.
Astrophysics studies Potekhin05  of carbon even at 100 eV have shown
discontinuities near $\bar{\rho}\sim 3$g/cm3 in several physical properties.

Figure 5: C-C pair potentials at densities in regions separated by the
discontinuities in the NPA pressure. The densities shown
($4,\,3.6,\,3.3,\,2.9$ g/cm3) are for regions where the coordination numbers
$N_{c}\sim 10,8,6,$ and $N_{C}<6$ prevail. The discontinuities imply LPTs in
the NPA calculation. The QMD data imply an LPT separating the $N_{c}=6$ and 8
regions.

Discussion – The $S(k)$ data have shown that the ionic structure of the fluid
is determined by strong electron scattering across the Fermi surface, and by
the Friedel oscillations of the pair-potentials. This ‘Coulomb fluid’ model of
$l$-carbon provides a complete account of the structural and thermodynamic
properties of $l$-carbon without invoking covalent bonds, using only simple LR
pair-potentials. The covalent bonds are mere transient bonds, lasting the
life-time of longitudinal phonons, while DFT deals with thermodynamic
($\omega\to 0$) averages.

The LPTs implied by the discontinuities in $P$ and in $\kappa$ seen in Fig. 3
in NPA occur near densities $\sim 3.15,\,3.54,\,3.75$ g/cm3 at 1 eV, with the
coordination number $N_{c}$ changing to 6, 8, and 10 respectively (see SM,
i.e., Appendix). The LPT3.5 near $\bar{\rho}_{\rm D},N_{c}=8\to 6$ is clearly
seen in the QMD-SCAN $P$ data that accurately fall on two straight lines
crossing at 3.42 g/cm3. The change in $N_{c}$ follows a change in the sign of
the NPA pseudopotential at the diamond density, when $U_{ei}(2k_{F})$ becomes
zero.

The three LPTs occur while the fluid conserves the structure of $S(k)$ near
2$K_{F}$ at each change of $N_{c}$, while $\bar{Z}$ also remains constant.
Hence the conductivity may be continuous across these LPTs. However, a slight
discontinuity is seen at the LPT3.5 as discussed in the SM (i.e., Appendix) SM
.

The minima in the pair potential $V(r)$ (Fig. 5) are all positive in the high
density region (e.g., $\bar{\rho}=4$ g/cm3), with $N_{c}\sim$ 10 or more. The
next lower density region develops a deeper first minimum and negative
secondary minima. The third region (e.g. $\bar{\rho}=3.3$g/cm3) pulls in ions
to the center by making the secondary minima less attractive. In the lowest
density region the first minimum becomes very deep, and eventually becomes
negative enough to form persistent covalent bonding.

The NPA calculation is for a uniform fluid where the 2$k_{F}$ scattering is
tightly linked to the liquid structure and the coordination number through the
Friedel minima in the pair potential that correlate the whole fluid. Density
fluctuations near a phase transition become very long ranged. Large-$N$ QMD
simulations may be necessary to recover such effects. If not, the sharp
transition gets ‘rounded off’, unless at least one phase is a crystalline
solid. It is hypothesized that $l$-carbon passes from a uniform fluid to a
higher-density uniform fluid at 3.5g/cm3 LPT which is a first-order
transition. The LPTs at 3.15 and 3.75 g/cm3 show discontinuities only in the
compressibility. They may involve narrow amorphous intermediates bridging
uniform phases, since the QMD pressure distributions at these LPTs fit to two
Gaussians.

In conclusion, the ionic and electronic structure of $l$-carbon can be
accurately modeled as that of a uniform liquid metal whose structure is
dominated by strong electron-ion interactions at the Fermi surface. We find
three liquid-liquid phase transition at 1 eV in the range 2.9-4.0 g/cm3
studied in detail here, as confirmed by both NPA and QMD calculations.

*

## Appendix A

Supplemental Material in support of the main text titled: A ‘simple metal’
description of liquid carbon, its warm-dense matter states, and the
identification of liquid-liquid phase transitions.

Here we use the same abbreviations, acronyms and units (Hartree atomic units)
as defined in the main text. This supplemental material (SM) deals with the
following topics.
(i) Details of the QMD-SCAN calculations used in this study
(ii) The Gaussian distributions for QMD-SCAN estimates of the equilibriated
pressure.
(iii)One-atom density functional theory used in the NPA
(iv) The linear-response (LR) potentials and the electron response function
used in the NPA method.
(v)Examination of the origin of discontinuities in the pressure.
(vi)The electrical conductive of liquid carbon.

Details of the QMD-SCAN calculations used here.
We have used finite-$T$ DFT-MD (QMD) calculations where classical molecular
dynamics is used to evolve 108 ions in a cubic simulation cell, while the
electrons, with four ionized electrons per carbon atom are treated quantum
mechanically using density functional theory. The numerical code implemented
in the Vienna ab initio simulation package VASP 5.4.4 VASP , and the projector
augmented wave pseudopotential for the interaction between the nuclei and the
electrons provided in VASP were used. The exchange and correlation potential
is approximated by the SCAN functional which has been found to perform better
for systems with covalent interactions Remsing17 . The simulations used an
energy cutoff of 414 eV and a simulation time of 4 ps. The ion temperature was
controlled with a Nosé thermostat, and enough empty bands were included to
ensure that the highest energy bands had negligible occupations. Evaluations
of the Brillouin zone were performed at the Baldereschi $k$-point Baldereshi73
.

The Gaussian distributions for QMD-SCAN estimates of the equilibrium pressure.
Since the evidence for the LPTs are to be based on breaks in the predicted
pressure, particular care was used to determine the equilibrium pressure from
the simulation in a consistent, non-subjective manner. The QMD simulation
records a distribution of pressures around the thermodynamic mean value, and
they form a Gaussian distribution for a uniform fluid at equilibrium. Hence
the simulation data were fitted to a Gaussian and the most probable pressure
was determined from the peak of the Gaussian. In two cases, viz., near the
LPT3 and LPT3.75, an average over two Gaussians seemed more appropriate. This
is illustrated in Figs. 6 and 7.

Figure 6: (Color online) (a) Pressure ‘readings’ at 1 eV from a DFT-MD
simulation using 108 atoms at 3.1 g/cm3 very close to the LPT3. The ‘run
number’ is an arbitrary index. (b) The pressure ‘readings’ are fitted to a
Gaussian (unnormalized), and the equilibrium pressure is taken to be that
corresponding to the maximum of the Gaussian. In the present case a bimodal
character is clear and a sum of two Gaussians with weights of 0.699 and 0.301
is appropriate, as in Fig. 4 of the main text.  Figure 7: (Color online) (a)
Pressure ‘distributions’ at 1 eV fitted to Gaussians. The distributions at
LPT3 and LPT${}_{3}.75$ use a two-Gaussian fit. Although there is a
discontinuity in the QMD-SCAN pressure at 3.42g/cm3, the $P$ distribution at
3.4g/cm3 shows no clear tendency to a bimodal distribution

The pressure evaluated in this manner, using the SCAN XC-functional is plotted
as SCAN-P in Fig. 3(b) of the main text. The width of the unnormalized
Gaussian distribution, $\Delta P$ is used to characterize the density
fluctuations in the system and displayed in panel (b) of Fig. 3 of the main
text, and labeled SCAN-$\Delta P$. Using the fit form
$f(p)=a\exp\\{-w^{2}(p-p_{m})^{2}\\}$ for the frequency of occurrence of the
pressure, $p_{m}$ is the mean pressure, while $w$ was taken as a measure of
the width of the distribution. The width has been multiplied by a factor of
3.3 in order to plot it in the same range as the other curves.

The one-atom density functional theory used in the NPA. Detailed discussions
of the NPA may be found in several recent publications Stanek21 ; cdwSi20 ,
besides some of the earlier publications DWP82 ; Pe-Be ; eos95 . In this
study, the density $\bar{\rho}$, and temperature $T$ (in energy units) are
such that the carbon atom is found to carry only the 1$s$ shell of bound
electrons, providing a very simple model of an atom in a plasma. Hence, for
simplicity of discussion we develop the two coupled DFT equations, i.e., for
electrons and for the ions that are solved in the NPA, in the following
simplified form where Hartree atomic units ($|e|=\hbar=m_{e}=1$) are employed.

In our ‘one-atom’ DFT model, coupled equations resulting from the stationary
condition of the grand potential $\Omega(n,\rho)$ considered as a functional
of the one body electron and ion densities $n(r),\rho(r)$ are solved. A single
carbon nucleus is taken as the origin, and immersed in the electron and ion
distributions of the fluid, forming the system to be considered. Initially,
suitable trial values of $n(r)$ and $\rho(r)$ are assumed. The coupled
equations are:

$\displaystyle\delta\Omega\left[n,\rho\right]/\delta n=0$ (3)
$\displaystyle\delta\Omega\left[n,\rho\right]/\delta\rho=0$ (4)

where the functional derivatives are implied. The first of these reduces to
the familiar Kohn-Sham equation for the electron density around the nucleus,
while the second reduces to a classical DFT equation for the ion distribution.

A bare carbon nucleus of charge $Z_{n}$ is the origin of coordinates of the
uniform system of electrons and ions. Spherical symmetry is applicable, as we
consider a uniform fluid. The inhomogeneous densities $n(r),\rho(r)$ around
the carbon nucleus become the ‘bulk’ densities $\bar{n},\bar{\rho}$ at large
distances $r\to R_{c}$, where $R_{c}$ is the about ten Wigner Seitz (WS)
radii. A shorter $R_{c}\sim 5r_{\rm ws}$ can be used at higher temperatures.
Here $r_{\rm ws}$ is given by $r_{\rm ws}=\\{3/(4\pi\bar{\rho})\\}^{1/3}$.
Thus, unlike in average-atom (AA) models where the electrons of an ion are
confined to the WS-sphere within which the Kohn-Sham equation is solved, the
NPA uses the large ‘correlation sphere’ of radius $R_{c}$ in solving the DFT
equations. The pair-distribution functions (PDFs) $g_{ab}(r),a=e,i$ refer to
electrons or ions and describe the structure of the environment where the
carbon atom is placed. Then

$\rho(r)=\bar{\rho}g_{ii}(r),\;n(r)=\bar{n}g_{ei}(r)$ (5)

The PDF $g_{ii}(r)$ will be referred to as $g(r)$ for brevity. The grand
potential can be written as

$\Omega=T[N,\rho]+\Omega_{e}+\Omega_{ei}+\Omega_{i}.$ (6)

Here $T[n,\rho]$ is the kinetic energy functional of a noninteracting system
having the exact interacting densities. A simplified form for $\Omega_{e}$ is
given below merely for ease of presentation, assuming a point ion-model
$U_{ei}(r)=-\bar{Z}/r$ for the electron-ion interaction. The ion contribution
$\Omega_{i}$ is also discussed below. The more complete model, applicable even
to ion mixtures is found in Refs. eos95 ; Pe-Be  and used in the computations
.

$\displaystyle\Omega_{e}$ $\displaystyle=$ $\displaystyle-\int d{\bf
r}\frac{Z_{n}}{r}n(r)+\frac{1}{2}\int d{\bf r}d{\bf
r^{\prime}}\frac{n(r)n(r^{\prime})}{|\bf{r}-\bf{r}^{\prime}|}$ (7)
$\displaystyle+\int d{\bf r}F^{xc}_{ee}[n]-\mu_{e}\int d\bf{r}n(r),$
$\displaystyle\Omega_{ei}$ $\displaystyle=$ $\displaystyle-\int d{\bf r}d{\bf
r^{\prime}}\frac{\bar{Z}\rho(r)n(r^{\prime})}{|\bf{r}-\bf{r}^{\prime}|}+\int
d{\bf r}F^{xc}_{ei}[n,\rho]$ (8)

Note that three-body and higher contributions beyond pair interactions are all
contained in the XC-functionals, and are not neglected in the theory. The one-
atom DFT approach (viz., the NPA) does not use an $N$-center potential energy
surface due to the ions for the Kohn-Sham electrons, as is the case with the
$N$-atom DFT deployed in VASP VASP  and similar codes. What we have is the
appropriate one-atom mapping of the $N$-atom DFT calculation.

We have used in Eq. 7 a point-ion model $-\bar{Z}/r$ for the electron-ion
interaction of the field ions only for simplicity of presentation. In actual
calculations a local pseudopotential $U_{ei}(q)=-\bar{Z}V_{q}M_{q}$, where
$V_{q}=4\pi/q^{2}$ and a form factor $M_{q}$ are used. The form factor, and
the corresponding ion-ion pair potential are also determined self-consistently
from the NPA, as discussed elsewhere Pe-Be ; eos95 ; Stanek21 . Non-local
forms of the pseudopotential have not been found necessary for NPA
calculations for uniform-density warm dense fluids or for cubic solids.

The ion contribution $\Omega_{i}$, though not displayed with the equations,
can be obtained from the above equations by appropriately replacing $n(r)$ by
$\bar{Z}\rho(r)$ if the ion-electron interaction is modeled by point ions,
while also replacing $F^{xc}$ contributions appropriately. The electron-ion
xc-functional $F^{xc}_{ei}$ is usually neglected in most NPA calculations,
being largely equivalent to making the Born-Oppenheimer approximation, and
neglecting certain correlation corrections of the from $\langle n({\bf
r})\rho({\bf r}^{\prime})\rangle-\langle
n(r)\rangle\langle\rho(r^{\prime})\rangle$. This is equivalent to using a
‘random-phase’ approximation for the electron-ion response function. This is
appropriate for dense uniform fluids of carbon studied here. However, such
correlations may be important at low densities, and with composite carbon
grains and amorphous systems.

The stationary condition on $\Omega$ under functional variation $\delta n$
leads to the usual Kohn-Sham (KS) equation for electrons moving in an
effective potential $U_{e}(r)$. Functional differentiation with respect to
$\delta\rho$ leads to an equation identifiable with the modified hyper-netted-
chain (MHNC) equation if the ion-ion XC-functional is identified with the
hyper-netted-chain (HNC) diagrams and bridge diagrams used for classical
systems. The ions are classical in the regime of study, and there is no
exchange contribution. Then the effective classical KS potential for the ions
can be identified with the ‘potential of mean force’, $U_{ii}(r)$ (see Sec.
III of Ref. DWP82 ) of classical statistical mechanics.

The bridge contributions can be included using the hard-sphere model, with the
hard sphere packing fraction parameter $\eta$ selected using the Lado-Foils-
Ashcroft criterion LFA83 . We find that the inclusion of bridge corrections
has only a negligible effect for $l$-carbon in the range of $\bar{\rho},T$
studied here. Typical values are given below

Table 1: . The pressure at 1 eV without (labeled no B) and with (labeled with B) bridge corrections, and the corresponding hard-sphere packing fraction $\eta$ for selected densities. $\rho$ g/cm3 | 3.0 | 4.0 | 8.0
---|---|---|---
$P$ TPa, no B | 0.03413 | 0.1705 | 1.540
$P$ TPa, with B | 0.03359 | 0.1700 | 1.538
$\eta$ | 0.1080 | 0.2030 | 0.3515

In solving the Kohn-Sham equation, the field ion distribution
$\rho(r)=\bar{\rho}g(r)$ occurring in $\Omega_{e}(r),\Omega_{ei}(r)$ as well
as in the corresponding KS equation $\delta\Omega/\delta n$ is replaced by
$\bar{\rho}g_{cav}(r)$, where $g_{cav}(r)$ is a model ion-ion PDF which is
just a spherical cavity of radius $r_{\rm ws}$. Hence solving the electron KS
equation coupled to the ion KS equation is much simplified, and the only
parameter associated with the ion distribution that has to be varied self-
consistently is the ion Wigner-Seitz radius $r_{\rm ws}$ appropriate to a
given free electron density $\bar{n}$ given as the input. Thus the primary
input variable is the free electron density, for a given temperature and
nuclear charge. The equilibrium ion density $\bar{\rho}$ is determined for
each given $\bar{n}$ in this manner, while solving the electron KS equation
self-consistently, starting from a trial $n(r)$ and $\bar{Z}$. At the end of
the calculation, the effect of the cavity distribution is subtracted off using
linear response theory. This provides us the nonlinear response (Kohn sham
electron density) of an interacting uniform electron fluid to the carbon ion
with its set of bound electrons.

The self-consistent solution for the continuum and bound state solutions is
constrained to satisfy the Friedel sum rule and verified for satisfying the
$f$-sum rule. The xc-functional used for the electron KS equation is the
finite-$T$ xc-functional of Perrot and Dharma-wardana PDWXC  within the local
density approximation (LDA). The xc-functional depends on $T/E_{F}$, where
$E_{F}$ is the Fermi energy of the free electrons. The electron system is a
strongly degenerate quantum gas in the system under study. A comparison of the
finite-$T$ xc-functional used here with the parametrization due to Dornheim et
al. Dornheim18  fitted to quantum Monte Carlo data showed good agreement cdw-
N-rep19 .

This partial decoupling of the electron Kohn-Sham equation and the ion Kohn-
Sham equation used in the NPA implementation is possible because the electron
Kohn-Sham equation is found to be only weakly dependent on the details of
$g_{ii}(r)$ for $r>r_{\rm ws}$. As already noted, we use the free electron
part of $n(r)$, viz., $n_{f}(r)=\bar{n}+\Delta n_{f}(r)$ obtained from the KS
equation to construct the $\Delta n_{f}(r)$ that would be obtained if there
were no $g_{cav}(r)$, using linear response (LR) theory. The corrected
$\Delta_{f}n(r)$ is the response of a uniform electron fluid (in the presence
of a non-responding neutralizing uniform ion background) to the carbon ion of
charge $\bar{Z}$. Hence this corrected $\Delta n_{f}(r)$ may be regarded as
being independent of the assumed form of $g(r)$ in solving the electron KS
equation, as long as it satisfied basic criteria in regard to charge
neutrality and the perfect screening sum rule.

Once the electron Kohn-Sham equation is solved using $g_{cav}(r)$, we already
have the three quantities $\bar{n},\bar{Z}$ and hence $\bar{\rho}$. We also
have the KS eigenfunctions $\phi_{\nu}(r)$ and eigenvalues $\epsilon_{\nu}$,
with $\nu=n,l$ for bound states, and $\nu=k,l$ for continuum states, with
$\epsilon_{k}=k^{2}/2$, together with the phase shifts $\delta_{kl}$. These
satisfy the Friedel sum rule and the simple charge neutrality condition
$\bar{n}=\bar{Z}\bar{\rho}$.

The linear response (LR) function of the electron fluid and its use in
calculating the pseudopotential. The KS calculation for the electron states
for the NPA in a fluid involves solving a simple radial equation. The
continuum states $\phi_{k,l}(r),\epsilon_{k}=k^{2}/2$, with occupation numbers
$f_{kl}$, are evaluated to a sufficiently large energy cutoff and for an
appropriate number of $l$-states (typically 9 to 39 were found sufficient for
the calculations presented here). The very high-$k$ contributions are included
by a Thomas-Fermi correction. This leads to an evaluation of the free-electron
density $n_{f}(r)$, and the free-electron density pileup $\Delta
n^{\prime}(r)=n_{f}(r)-\bar{n}$. A part of this pileup is due to the presence
of the cavity potential. This contribution $m(r)$ is evaluated using its
linear response to the electron gas of density $\bar{n}$ using the interacting
electron response $\chi(q,T_{e})$. The cavity corrected free-electron pileup
$\Delta n_{f}(r)=\Delta n^{\prime}(r)-m(r)$ is used in constructing the
electron-ion pseudopotential as well as the ion-ion pair potential $V_{ii}(r)$
according to the following equations (in Hartree atomic units), given for
Fourier-transformed quantities:

$\displaystyle U_{ei}(k)$ $\displaystyle=$ $\displaystyle\Delta
n_{f}(k)/\chi(k,T_{e}),$ (9) $\displaystyle\chi(k,T_{e})$ $\displaystyle=$
$\displaystyle\frac{\chi_{0}(k,T_{e})}{1-V_{k}(1-G_{k})\chi_{0}(k,T_{e})},$
(10) $\displaystyle G_{k}$ $\displaystyle=$
$\displaystyle(1-\kappa_{0}/\kappa)(k/k_{\text{TF}});\quad V_{k}=4\pi/k^{2},$
(11) $\displaystyle k_{\text{TF}}$ $\displaystyle=$
$\displaystyle\\{4/(\pi\alpha r_{s})\\}^{1/2};\quad\alpha=(4/9\pi)^{1/3},$
(12) $\displaystyle V_{ii}(k)$ $\displaystyle=$ $\displaystyle
Z^{2}V_{k}+|U_{ei}(k)|^{2}\chi_{ee}(k,T_{e}).$ (13)

Here $\chi_{0}$ is the finite-$T$ Lindhard function, $V_{k}$ is the bare
Coulomb potential, and $G_{k}$ is a local-field correction (LFC). The
finite-$T$ compressibility sum rule for electrons is satisfied since
$\kappa_{0}$ and $\kappa$ are the non-interacting and interacting electron
compressibilities respectively, with $\kappa$ matched to the $F_{xc}(T)$ used
in the KS calculation. In Eq. 12, $k_{\text{TF}}$ appearing in the LFC is the
Thomas-Fermi wavevector. We use a $G_{k}$ evaluated at $k\to 0$ for all $k$
instead of the more general $k$-dependent form (e.g., Eq. 50 in Ref. PDWXC )
since the $k$-dispersion in $G_{k}$ has negligible effect for the WDMs of this
study. The xc-functional is used in the LDA which is efficient and accurate
because the one-center electron density $n(r)$ is smooth compared to the
complex $N$-center electron density used in VASP-type $N$-center DFT
calculations.

Examination of the origin of discontinuities in the pressure.
It is known that average-atom (AA) models Murillo13  that confine all the
bound and free electrons associated with an ion in the Wigner Seitz sphere can
produce spurious discontinuities in the pressure and in the mean ionization
$\bar{Z}$. In the NPA the elctrons are not confined to the WS-sphere, but to
the correlation sphere of radius $R_{c}\sim 10r_{\rm ws}$-$5r_{\rm ws}$.
Average atom models have to deal with electrons that “leak out” of the WS-
sphere, and model dependent effects arising from the choice of boundary
conditons at the surface of the WS-sphere; these do not arise in the NPA.

However, NPA models as well as AA models have to deal with discontinuities
that arise when a well-confined bound state moves upwards in energy and into
the continuum due to changes in density or temperature. In some cases, such
ionization is accompanied by phase transitons and discontinuities in physical
properties. In other cases, when unphysical discontinuities occur, they can be
corrected by ensuring that appropriate discontinuties that cancel them are
properly included in the xc-functionals. However, we have no changes in the
degree of ionization ($\bar{Z}$) or any other crtical parameters in the range
of densities studied here, and hence no spurious discontinuities are expected
in this range of $\bar{\rho},T$.

The NPA pressure is obtained from an evaluation of the Helmholtz free $F$ of
the system. This consists of contributions of the form Pe-Be ; eos95 :

$F=F^{0}_{e}+F^{xc}_{e}+F^{a}_{em}+F^{b}_{em}+F_{12}+F^{0}_{I}.$ (14)

The first two terms deal with the free energy of the non-interacting uniform
electron fluid and its finite-$T$ exchange-correlation energy at the given
density and temperature $T$. The last term, $F^{0}_{I}$ is the ideal
(classical) free energy of the ion subsystem. The third and fourth terms
together form the embedding energy of the nucleus and the inhomogeneous
electron density that form neutral pseudo-atom in the uniform electron fluid.
The fifth term, $F_{12}$ contains the interactions between pseudoatoms brought
in via the pair-potential, pair-distribution functions and the ion-ion
correlation effects. The contribution to the pressure from all the terms
except $F_{12}$ can be expressed analytically. The density derivative of the
term $F_{12}$ has to be evaluated numerically. Then the pressure can be
written as:

$P=P^{0}_{e}+P^{ex}_{e}+P^{a}_{em}+P^{b}_{em}+P_{12}+P^{0}_{I}.$ (15)

No discontinuities are expected in $P^{0}_{e},P^{ex}$ or $P^{0}_{I}$ when
treated as a function of $\bar{\rho}$ as the number of free electrons per ion,
$\bar{Z}$ remains constant at a value of four in this density range. Hence, in
Fig. 8, panel (a) we display only the two components of the embedding energy
and the pair-interaction free energy $F_{12}$ as a function of the density, at
1 eV.

Figure 8: (Color online) (a) We display the two components of the embedding
pressure obtained without numerical differentiation from the embedding free
energy of the neutral pseudo atom. The pair-interaction free energy $F_{12}$
which contains bonding effects and many-ion correlation effects is also
displayed. The embedding pressure component $P^{b}_{em}$ displays three
discontinuities, while $F_{12}$ shows a slight discontinuity near
$\bar{\rho}=3.52$. (b) We display the $k\to 0$ limit of the structure factor
which is $S(0)=\kappa/\kappa_{0}$, where $\kappa_{0},\kappa$ are the ideal
fluid compressibility of the ions, and the interacting compressibility
respectively. The value of $S(0)$ at 10 eV is shifted by 0.15 for ease of
comparison, and shows no discontinuities. Similarly, the pressure at 10 eV,
displayed in the Fig.3(a) of the main text, shows no discontinuities.

In order to understand the physical content of the embedding pressure (Pe-Be )
we write down the potential that produces a single neutral pseudo atom in the
uniform fluid of mean electron density $\bar{n}$ and mean ion density
$\bar{\rho}$, viz.,

$V^{npa}=\frac{1}{r}\star(-\bar{Z}\delta_{0}+\nu+\Delta n).$ (16)

Here $\bar{Z}\delta_{0}$ defines the nuclear term at the origin, while
$\nu(r)$ is the density of the spherical cavity which mimics the ion
distribution $\rho(r)$ by $\bar{Z}g_{cav}(r)$.

$\nu(r)=\bar{n},\,r>r_{\rm ws},\mbox{ else }\nu(r)=0.$ (17)

Also, $\Delta n$ is the displaced electron density with reference to the mean
electron density $\bar{n}$. The symbol $\star$ defines a convolution product.
We also define the integration over all space via the symbol $\circ$.

$\displaystyle f\star g$ $\displaystyle=$ $\displaystyle\int
f(\vec{r})g(\vec{r}-\vec{s})d\vec{s}$ (18) $\displaystyle f\circ g$
$\displaystyle=$ $\displaystyle\int f(r)g(r)d\vec{r}.$ (19)

The volume are over a sphere of radius $R_{c}\sim 10r_{\rm ws}$. Then,
denoting the volume of the Wigner-Seitz sphere by $\Omega_{\rm ws}$, the two
embedding pressure terms can be written as Pe-Be :

$\displaystyle P^{a}_{em}$ $\displaystyle=$ $\displaystyle(\nu-\bar{n})\circ
V^{npa}/\Omega_{\rm ws}$ (20) $\displaystyle P^{b}_{em}$ $\displaystyle=$
$\displaystyle-\bar{Z}V^{npa}(r_{\rm ws}).$ (21)

Since $\bar{Z}=4$, and remains at that value through out the range of
densities studied, the discontinuities in the embedding pressure are caused
solely by the ‘external potential’ associated with the creation of the carbon
pseudoatom, and not associated with bonding with other carbon ions (such
contributions are in $F_{12}$).

This re-enforces our conclusion that the observed LPTs are associated with
changes in coordination number $N_{c}$ and the effect on the electron-ion
interactions at the Fermi energy via $q=2k_{F}$ scattering. The coordination
number $N_{c}$ can be approximately determined from the area under the first
peak of $g(r)$, and has been reported in detail by Vorberger et al Vorberger20
. They do not cause discontinuities is $P$ but clearly cause discontinuites in
the compressibility (as seen in $S90)$, i.e., a derivative of the pressure.

The electrical conductivity of liquid carbon.
The static electrical conductivity $\sigma$ usually changes at phase
transitions because at least one of several physical parameters change
abruptly. These are the number of free carriers, the distribution of the ions
that cause the scattering of electrons, and the scattering potential. These
properties are contained in the pseudopotential $U_{ei}(k)$, the structure
factor $S(k)$ and the screening function of the system, when considering the
static conductivity.

In trhe present case $\bar{Z}$ remains unchanged at four. The high value of
the Fermi energy $E_{F}$ of liquid carbon, e.g., 28.8 eV near the diamond
density of 3.5 g/cm3, implies that at 1 eV the electrons are nearly completely
degenerate, and hence electron scattering can occur only from one edge of the
Fermi surface to the other, with a momentum transfer $q=2k_{F}$, where $k_{F}$
is the Fermi momentum. Thus the quantities $U_{ei}(2k_{F}),S(2k_{F})$ and the
inverse dielectric function at $2k_{F}$ which screens the pseudopotential
determine the electrical conductivity. We noted that, due to the strong
electron-ion scattering at 2$k_{F}$, the structure factor remains tied to a
peak at 2$k_{F}$. Thus sharp changes are not expected in the electrical
conductivity of this system under the liquid-liquid phase transitions.
However, an extremely weak discontinuity can be seen (Fig. 9) at the LPT3.5,
i.e., at the nominal diamond density where we also noted a discontinuity in
the pressure calculated via QMD-SCAN, and via the NPA model. Given the very
small magnitude of the discontinuity in $\sigma$, it is not surprising that no
discontinuities in $\sigma$ are seen at LPT3 and LPT3.75 where the pressure
isotherm also did not show any discontinuities.

Figure 9: (Color online) The conductivity calculated using the NPA
pseudopotential $U_{ei}(k)$ and the structure factor $S(k)$ in the Ziman
formula is displayed. The conductivity results reported by Vorberger et al. is
also displayed. The slight discontinuity in $\sigma$ at $\simeq 3.53$g/cm3 is
shown in more detail in the inset. The discontinuity coincides with the
LPT3.5. The conductivity shows no discernible discontinuities at the other two
LPTs.

The QMD estimate of the conductivity of liquid carbon is approximately half
that of the NPA-Ziman estimate. This difference becomes even larger at higher
densities, and may be associated with the fact that the NPA model assumes a
strictly homogeneous fluid, while the QMD simulation considers the fluid to be
an average over many realizations of a large variety of inhomogeneous atomic
clusters. While such averaging is effective when the ions themselves actually
easily form a uniform system, it is not evidently so in systems where there is
transient bonding as well as incipient formation of preferred clusters
dictated by the tetrahedral character of the fluid. Hence, when such systems
form homogeneous fluids, the QMD simulation may need a large number of atoms.
On the other hand, if the fluid is truly a dispersed phased of atomic
clusters, the NPA approach that assumes a uniform-density fluid cannot
accurately describe such a system using the simple ion-ion xc-functionals
based on the Ornstein-Zernike and the modified HNC approach (i.e., HNC+bridge)
used here.

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