# Generalized Bassian and Other Mixed
Abelian Groups With Bounded $p$-Torsion

Peter V. Danchev  Institute of Mathematics and Informatics, Bulgarian Academy
of Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Patrick W. Keef  Department of Mathematics, Whitman College, Walla Walla,
WA 99362, USA<EMAIL_ADDRESS>
###### Abstract.

It is known that a mixed abelian group $G$ with torsion $T$ is Bassian if, and
only if, it has finite torsion-free rank and has finite $p$-torsion (i.e.,
each $T_{p}$ is finite). It is also known that if $G$ is generalized Bassian,
then each $pT_{p}$ is finite, so that $G$ has bounded $p$-torsion. To further
describe the generalized Bassian groups, we start by characterizing the groups
in some important classes of mixed groups with bounded $p$-torsion (e.g., the
balanced-projective groups and the Warfield groups). We then prove that all
generalized Bassian groups must have finite torsion-free rank, thus answering
a question recently posed in Acta Math. Hung. (2022) by Chekhlov-Danchev-
Goldsmith. This implies that every generalized Bassian group must be a
B+E-group; i.e., the direct sum of a Bassian group and an elementary group.
The converse is shown to hold for a large class of mixed groups, including the
Warfield groups. It is also proved that $G$ is a B+E-group if, and only if, it
is a subgroup of a generalized Bassian group.

###### Key words and phrases:

Abelian groups, Bassian groups, Generalized Bassian groups, Balanced-
Projective groups, Warfield groups

###### 2010 Mathematics Subject Classification:

20K10

##  1\. Introduction

All groups considered will be abelian and additively written. Our notation and
terminology will mainly agree with the classical book [4]. For instance, if
$G$ is a group, the letter $T=T(G)$ will denote its maximal torsion subgroup
with $p$-primary component $T_{p}=T_{p}(G)$. If $x\in G$, then $|x|_{p}$ will
denote its $p$-height. A torsion group $T$ is elementary if each $T_{p}$ is
$p$-bounded. Clearly, any subgroup of an elementary group is a summand.

The following notions are central to our investigations.

######  Definition 1.1.

([2],[3]) A group $G$ is said to be Bassian if it cannot be embedded in a
proper homomorphic image of itself or, equivalently, if the existence of an
injection $G\to G/N$, for some subgroup $N$ of $G$, forces that $N=\\{0\\}$.

More generally, if this injective homomorphism implies that $N$ is a direct
summand of $G$, then $G$ is called generalized Bassian.

Restated slightly, the Main Theorem of [2] is that $G$ is Bassian if, and only
if, it has finite torsion-free rank and, for all primes $p$, $T_{p}$ is also
finite. The generalized Bassian groups were not fully characterized in [3],
but it was shown that if $G$ is generalized Bassian, then $pT_{p}$ must be
finite for all primes $p$ ([3], Lemma 3.1). The main goal of this paper is to
describe the class of generalized Bassian groups in greater detail. In
pursuing this goal we will discuss a number of results applicable to any mixed
group with bounded $p$-torsion (i.e., all $T_{p}$ are bounded). Concretely,
our work is organized thus:

In Section 2 we introduce some examples of groups with bounded $p$-torsion.
Two of the most important classes of mixed abelian groups are the Warfield
groups, and its subclass consisting of the balanced-projective groups,
introduced in [10] and [6], respectively (we define these classes later; see
also [7] and [8]). We completely characterize the groups in these classes
which have bounded $p$-torsion (Propositions 2.3 and 2.5).

In Section 3 we prove that, as with Bassian groups, any generalized Bassian
group necessarily has finite torsion-free rank (Theorem 3.5). This gives an
affirmative answer to a question from [3] (Section 4). Second, we show that
any group $G$ with finite torsion-free rank and bounded $p$-torsion must
necessarily be isomorphic to a direct sum $H\oplus S$, where $H$ is a Bassian
group and $S$ is torsion (Theorem 3.1). This gives much more direct connection
between the classes of Bassian and generalized Bassian than was apparent in
[3].

The following definition is important:

######  Definition 1.2.

A group $G$ is called B+E (or alternatively, is said to be a B+E-group) if it
decomposes as $G=A\oplus E$, where $A$ is Bassian and $E$ is elementary.

We show that $G$ is B+E if, and only if, it has finite torsion-free rank and
for all primes $p$, $pT_{p}$ is finite (Proposition 3.3), which implies that
any subgroup of a B+E-group shares that property (Corollary 3.4). It also
follows that every generalized Bassian group is B+E (Corollary 3.6). The
converse of this implication is the focus of the remainder of the paper.

######  Conjecture 1.3.

A group is B+E if, and only if, it is generalized Bassian; i.e., the converse
of Corollary 3.6 also holds.

We prove that the class of B+E-groups for which the conjecture holds is quite
large and includes all those mentioned in Section 2. An important step in this
investigation is the observation that if $G=A\oplus E$ is B+E with $E$
elementary and $A$ has no infinite elementary summands, then $G$ is
generalized Bassian (Theorem 3.8).

In Section 4 we prove that many B+E-groups have a decomposition as that in
Theorem 3.8. This includes the Warfield groups (and hence the balanced-
projectives). Even more generally, we show that if $G$ is a B+E-group with a
nice free subgroup $F$ such that $G/F$ is torsion (i.e., $F$ is quasi-
essential in $G$), then $G$ is generalized Bassian (Theorem 4.7). This, in
turn, allows us to conclude that $G$ is a B+E-group if, and only if, it is a
subgroup of a generalized Bassian group (Corollary 4.10).

##  2\. Groups with bounded $p$-torsion

We begin by reviewing some terminology. Suppose $G$ is a group with torsion
subgroup $T$. Recall, $G$ has bounded $p$-torsion if $T_{p}$ is bounded for
all primes $p$. So, a generalized Bassian group (and hence a Bassian group)
has bounded $p$-torsion; in fact, each $pT_{p}$ (respectively, $T_{p}$) must
be finite.

We will use some basic ideas and terminology from the theory of valuated
groups (see, for example, [9]). Our valuated groups will have a valuation for
every prime $p$, which generalizes the $p$-height on a group.

For a valuated group $V$, prime $p$ and ordinal $\alpha$, let
$V(\alpha)=\\{x\in V:|x|_{p}\geq\alpha\\}$ and

$U_{V}^{p}(\alpha)=\\{x\in V(\alpha):|px|_{p}\geq\alpha+2\\}/V(\alpha+1)$

be the $\alpha$th Ulm invariant of $V$ and $f_{V}^{p}(\alpha)$ be the
corresponding value of the $p$-Ulm function of $V$.

If $V$ is a valuated group, then by a realization of $V$ we mean a group $G$
containing $V$ as a subgroup such that the $p$-valuations on $V$ agree with
the $p$-heights inherited from $G$. Note that, in this case, for every prime
$p$ and ordinal $\alpha$, there is a natural injection $U_{V}^{p}(\alpha)\to
U_{G}^{p}(\alpha)$ whose co-kernels can be identified with the relative Ulm
invariants of $V$ in $G$. We will primarily be concerned with nice
realizations, i.e., where $V$ is also a nice subgroup of $G$.

If $G$ is a group, or $V$ is a valuated group, and $p$ is a prime, we will
denote their localizations by $G_{p}$ and $V_{p}$, respectively; so,
$G_{p}=G\otimes\mathbb{Z}_{(p)}$ and $V_{p}=V\otimes\mathbb{Z}_{(p)}$. In
particular, this is consistent with the notation $T_{p}$ for the $p$-torsion
subgroup of $G$. There is a $p$-height preserving natural map $G\to G_{p}$
whose kernel is $\oplus_{q\neq p}T_{q}$.

Suppose $G$ is a realization of the free valuated quasi-essential subgroup
$F$. We will say that $F$ is tight if for all primes $p$, $U^{p}_{F}(j)\to
U^{p}_{G}(j)$ is an isomorphism for all $j<\omega$. Similarly, we will say $F$
is 0-tight if for all primes $p$, $U^{p}_{F}(0)\to U^{p}_{G}(0)$ is an
isomorphism.

A quasi-essential free subgroup $F$ of a group $G$ will be called a
decomposition subgroup if it is generated by a decomposition basis for $G$.
That is, $F$ is nice in $G$ and it is the valuated direct sum of a collection
of cyclic summands. So $G$ is a Warfield group (i.e., a summand of a simply
presented group) exactly when it has a decomposition subgroup such that the
factor-group $G/F$ is totally projective (where we are allowing divisible
torsion groups to be included as totally projective).

Note that if $G$ has bounded $p$-torsion, then $T_{p}$ must, as a pure bounded
subgroup of $G$, be a direct summand; i.e., $G=T_{p}\oplus A_{p}$ for some
$A_{p}$. If $x\in T_{p}$, then clearly $|x|$ is finite or $\infty$. Similarly,
if $x\in A_{p}$, then since multiplication by $p$ is an injective homomorphism
$A_{p}\to A_{p}$, it still follows that $|x|$ is finite or $\infty$.
Therefore, the same holds for any $x\in G$.

We will apply the following (essentially well-known) lemma to the case of
Warfield groups. Later, we prove a similar result for a broader class of
valuated groups (see Lemma 4.6 below).

######  Lemma 2.1.

Any infinite cyclic valuated group $X=\langle x\rangle$ such that, for all
primes $p$ and $k<\omega$, $|p^{k}x|_{p}$ is either finite or $\infty$, has a
tight realization $H$ that is simply presented of torsion-free rank $1$.

###### Proof.

For each prime $p$, consider the set $K_{p}\subseteq\omega$ of all $k$ such
that either $k=0$ or $|p^{k}x|\neq|p^{k-1}x|_{p}+1$ (where $\infty=\infty+1$).
For each prime $p$ and $k\in K_{p}$, define a set of generators, $Y_{p}^{k}$,
and relations as follows: If $|p^{k}x|=\infty$, let
$Y_{p}^{k}=\\{y_{p,j}^{k}\\}_{j<\omega}$ with relations $y_{p,0}^{k}=p^{k}x$
and $py_{p,j+1}^{k}=y_{p,j}^{k}$. On the other hand, if
$0\neq|p^{k}x|=n<\omega$, let $Y_{p}^{k}=\\{y_{p,j}^{k}\\}_{j\leq n}$ with
relation $y_{p,0}^{k}=p^{k}x$ and $py_{p,j+1}^{k}=y_{p,j}^{k}$

If $H$ is the group generated $x$ and the union of all the $Y_{p}^{k}$ with
the above relations, then $H$ is clearly simply presented, and a standard
argument shows that it is a realization of $X$. To show that $X$ is tight in
$H$, suppose $p$ is some fixed prime. It is fairly straightforward (though
somewhat tedious) to observe that if $T$ is the torsion subgroup of $H$, then
$T[p]$ is a free valuated vector space with a basis generated by the images of
all $p^{k-1}x-y_{p,1}^{k}$ such that $0<k\in K_{p}$, and that, similarly, the
non-zero Ulm invariants of $X$ are generated by the images of all such
$p^{k-1}x$, showing that $U_{X}^{p}(j)\to U_{H}^{p}(j)$ is always an
isomorphism, as promised. ∎

Though it could have been verified directly, in the last result, since $H$ has
torsion-free rank $1$, the subgroup $X$ must be nice in $H$.

######  Proposition 2.2.

Suppose $G$ is a Warfield group with bounded $p$-torsion. If $F$ is a
decomposition subgroup of $G$, then $G=H\oplus S$, where $H$ is a nice tight
simply presented realization of $F$ and $S$ is torsion.

###### Proof.

Let $\\{x_{i}\\}_{i\in I}$ be a decomposition basis for $F$. For each $i\in
I$, $X_{i}:=\langle x_{i}\rangle$ clearly satisfies the hypotheses of Lemma
2.1; let $H_{i}$ be a rank 1 simply presented tight realization of $X_{i}$ and
$H$ be the (external) direct sum of the $H_{i}$. Construct a direct sum of
cyclics $S$ such that for all primes $p$ and $j<\omega$, we have
$f^{p}_{F}(j)+f^{p}_{S}(j)=f^{p}_{G}(j)$. It follows that the relative Ulm
invariants of $F$ in $G$ agree with those of $F$ in $A\oplus S$ (both agree
with $f_{S}^{p}(j)$ for all $p$ and $j$). Therefore, the identity map $F\to F$
extends to an isomorphism $G\to H\oplus S$, as required. ∎

The following describes all Warfield groups with bounded $p$-torsion.

######  Proposition 2.3.

Suppose $G$ is a group with bounded $p$-torsion. Then the following are
equivalent:

(a) $G$ is simply presented;

(b) $G$ is the direct sum of a set of groups of torsion-free rank at most 1;

(c) $G$ is a Warfield group.

###### Proof.

That (a) implies (b) is well-known.

To show (b) implies (c), it is clear that if $G$ is torsion, then it is a
direct sum of cyclic groups and hence simply presented. So, assume $G$ has
torsion-free rank 1. Let $x\in G$ have infinite order. If $X=\langle x\rangle$
then it is well known that $X$ is nice in $G$. For each prime $p$, we claim
that $(G/X)_{p}$ is isomorphic to $B\oplus D$, were $B$ is bounded and $D$ is
either $0$ or $\mathbb{Z}_{p^{\infty}}$. This will imply that $G/X$ will be
totally projective, so that $G$ is Warfield.

To verify the claim, let $p$ be a prime and consider the localizations $X_{p}$
and $G_{p}$; so $G_{p}/X_{p}$ is isomorphic to $(G/X)_{p}$. Note that $T_{p}$
is a summand of $G_{p}$; suppose $G_{p}=T_{p}\oplus Y_{p}$. Since $Y_{p}$ is
either isomorphic to $\mathbb{Z}_{(p)}$ or $\mathbb{Q}$, it is readily checked
that in the first case $(G/X)_{p}$ is bounded and in the second case, it is
isomorphic to the direct sum of a bounded group and a copy of
$\mathbb{Z}_{p^{\infty}}$.

Finally, (c) implies (a) by Proposition 2.2. ∎

The following standard criterion is a direct consequence of Proposition 2.3.

######  Corollary 2.4.

A torsion-free group is Warfield if, and only if, it is completely
decomposable.

We now describe the balanced-projective groups with bounded $p$-torsion.
Imitating [10], recall that a generalized height $h$ means the formal direct
product $\prod_{p\in\mathbb{P}}p^{v(p)}$ (where $\mathbb{P}$ is the set of
primes), and if $G$ is a group, then
$h(G):=\bigcap_{p\in\mathbb{P}}p^{v(p)}G$, where $v(p)$ is either an ordinal
or the symbol $\infty$. A short-exact sequence

$0\to A\to B\to C\to 0$

is termed balanced if, for every generalized height $h$, the induced sequence

$0\to h(A)\to h(B)\to h(C)\to 0$

is exact. A group $P$ is said to be balanced-projective if it is projective
with respect to all such sequences, that is, for any such sequence, the
induced map $\mathrm{Hom}(P,B)\to\mathrm{Hom}(P,C)$ is surjective.

The following assertion describes all balanced-projective groups with bounded
$p$-torsion.

######  Proposition 2.5.

Suppose $G$ is a group with bounded $p$-torsion. Then the following are
equivalent:

(a) $G$ is balanced-projective;

(b) $G$ is a splitting mixed Warfield group;

(c) $G\cong T\oplus A$, where $A$ is a completely decomposable torsion-free
group.

###### Proof.

Suppose that (a) holds, i.e., $G$ is balanced-projective. It is well known
that $G$ must be a Warfield group. Note that, for every torsion-free element
$x\in G$, the coordinates of $h(x)$ are integers or $\infty$. It follows that
there is a torsion-free group of rank $1$, say $Q_{x}$, with $a\in Q_{x}$ and
$h(a)=h(x)$. The assignment $a\mapsto x$ defines a homomorphism $Q_{x}\to G$.
These determine a homomorphism

$\overline{G}:=T\oplus(\oplus_{x\in{G\setminus T}}Q_{x})\to G,$

which is clearly surjective and balanced-exact. Thus, it follows that $G$ is
isomorphic to a summand of $\overline{G}$ containing $T$, so that $G$ is
necessarily a splitting mixed group, as stated.

That (b) implies (c) follows directly from Corollary 2.4. Finally, since
(torsion) cyclic groups and rank-1 torsion-free groups are clearly balanced-
projective, (c) implies (a), completing the proof. ∎

The following example is crucial.

######  Example 2.6.

(a) There is a Bassian group that is not Warfield.

(b) There is a Bassian Warfield group that is not balanced-projective.

###### Proof.

For (a), we just consider any torsion-free group of finite rank that is not
completely decomposable. For example, any pure subgroup of the $p$-adic
integers, $J_{p}$, of finite rank exceeding 1 will satisfy the requirements.
For (b), let $G$ be generated by $t$ together with $s_{p}$ for each prime $p$,
subject to the relations $p^{2}s_{p}=pt$. Clearly, $G$ is simply presented,
and hence Warfield. Since $G$ has rank 1 and $T_{p}\cong\mathbb{Z}_{p}$ for
each prime, $G$ is Bassian. However, it is easy to check that $[t]\in G/T$ has
$h([t])=(1,1,1,\dots)$ where as, for any torsion-free element $x\in G$,
$|x|_{p}=0$ for all but finitely many primes $p$. It follows that $T$ is not a
summand of $G$, so that $G$ is not balanced-projective, as wanted. ∎

Suppose $T$ is a torsion group with bounded $p$-torsion. Note that
$\prod_{p}T_{p}/T$ is always torsion-free divisible. Suppose $G$ is a pure
subgroup of $\prod_{p}T_{p}$ containing $T$. For want of a better term, we
will call such a group PSP or just a PSP-group for “pure subgroup of the
product”. Alternately, if $G$ has bounded $p$-torsion, then $G$ is PSP if, and
only if, $G$ is Hausdorff in the $\mathbb{Z}$-adic topology and $T$ is dense
in $G$ in that topology. This follows since $\prod_{p}T_{p}$ will be the
completion of $T$ in the $\mathbb{Z}$-adic topology.

######  Proposition 2.7.

A PSP-group $G$ is Warfield if, and only if, there are decompositions
$T=\oplus_{i\in I}T^{i}$ and $G=\oplus_{i\in I}G^{i}$ such that, for each
$i\in I$, $G^{i}$ is a PSP-group of rank $1$ and torsion subgroup $T^{i}$.

###### Proof.

Certainly, if $G$ is Warfield, then by Theorem 2.3, $G$ must decompose into
groups of torsion-free rank $1$, as indicated. Clearly, each $G_{i}$ will be
Hausdorff in the $\mathbb{Z}$-adic topology with $T_{i}$ as a dense subgroup.
Therefore, each $G_{i}$ will also be a PSP-group.

The converse is an immediate consequence of Theorem 2.3. ∎

######  Example 2.8.

There is a Bassian PSP-group of rank $2$ that is not Warfield.

###### Proof.

Let $(a_{j},b_{j})$ for $j\in\mathbb{N}$ be an enumeration of
$\mathbb{Z}^{2}\setminus\\{(0,0)\\}$. For each $j\in\mathbb{N}$, we can
clearly find $x_{j},y_{j}\in\mathbb{Z}$ and a prime $p_{j}>p_{j-1}$ such
$p_{j}>|a_{j}x_{j}+b_{j}y_{j}|\neq 0$.

Let $T=\oplus_{j\in\mathbb{N}}\mathbb{Z}_{p_{j}}$ and $x:=[x_{j}]$,
$y:=[y_{j}]$ in $P:=\prod_{j\in\mathbb{N}}\mathbb{Z}_{p_{j}}$. If $a_{j}$ and
$b_{j}$ are integers, not both 0, and we consider the corresponding linear
combination $a_{j}x+b_{j}y\in P$, then the entire point of our construction is
that such a sum can be 0 for only a finite number of coordinates (those for
$i<j$). In particular, $x$ and $y$ are linearly independent over $T$ and there
is a PSP- group of torsion-free rank $2$ containing both.

If there were decompositions $T=T^{1}\oplus T^{2}$ and $G=G^{1}\oplus G^{2}$
as in Proposition 2.7, then clearly both $T^{1}$ and $T^{2}$ must have an
infinite number of non-zero components. Therefore, no linear combination of
$x$ and $y$ can be a non-zero element of either summand, which is clearly a
contradiction.

Therefore, $G$ is a Bassian PSP-group, but not Warfield, as claimed. ∎

##  3\. Groups with bounded $p$-torsion and finite torsion-free rank

We begin by characterizing the groups described in the title to this section.

######  Theorem 3.1.

The group $G$ has finite torsion-free rank and bounded $p$-torsion if, and
only if, is decomposable as $G=H\oplus S$, where $H$ is Bassian and $S$ is a
torsion group with bounded $p$-torsion.

###### Proof.

If $G=H\oplus S$ as above, it is easy to check that $r_{0}(G)=r_{0}(H)$ is
finite and $G$ has bounded $p$-torsion. So, assume $G$ satisfies these two
conditions.

Let $D$ be a divisible hull of $G/T$, so $D$ is a direct sum of a finite
number of copies of $\mathbb{Q}$. Since the map

${\rm Ext}(D,T)\to{\rm Ext}(G/T,T)$

is surjective, there is a group $G^{\prime}$ containing $G$ such that
$G^{\prime}/T\cong D$. If there is a similar decomposition
$G^{\prime}=H^{\prime}\oplus S^{\prime}$ for $G^{\prime}$, it readily follows
that $G^{\prime}$ also has torsion $T$ and we can simply let $S=S^{\prime}$
and $H=H^{\prime}\cap G$. So, there is no loss of generality in assuming that
$G/T=D$ is divisible.

Let $P=\prod_{p\in\mathbb{P}}T_{p}$; so we can think of $T$ as a subgroup of
$P$ and $Q:=P/T$ is torsion-free divisible. Now, the short-exact sequence

$0\to T\to P\to Q\to 0$

determines a long-exact sequence, which determines an isomorphism, as follows:

$[{\rm Hom}(D,P)=]\ \ 0\to{\rm Hom}(D,Q)\to{\rm Ext}(D,T)\to{\rm
Ext}(D,P)\to\cdots$

Note that

${\rm Ext}(D,P)={\rm Pext}(D,P)\cong\prod_{p\in\mathbb{P}}{\rm
Pext}(D,T_{p})=0,$

since $T_{p}$ is bounded, and hence pure-injective. Therefore, ${\rm
Hom}(D,Q)\to{\rm Ext}(D,T)$ is an isomorphism.

We can think of the extension

$\xi:0\to T\to G\to D\to 0$

as an element of ${\rm Ext}(D,T)$; in fact, suppose $\phi:D\to Q$ is the
corresponding element of ${\rm Hom}(D,Q).$ Let $y_{1},\dots,y_{n}$ be a
maximal linearly independent subset of $D$. So, $\phi$ is completely
determined by the $n$ elements $\phi(y_{1}),\dots,\phi(y_{n})$ of $Q$. For
$j=1,\dots,n$, choose $x_{j}\in P$ so that $\phi(y_{j})=x_{j}+T$. Note that
for each prime $p$, the $T_{p}$ coordinates from all of the $x_{j}$ can be
chosen to come from a single finite summand of $T_{p}$. In other words, for
each prime $p$ there is a decomposition $T_{p}=F_{p}\oplus S_{p}$, where
$F_{p}$ is finite, such that if $F=\oplus F_{p}$ and $P_{F}=\prod F_{p}$, then
$x_{j}\in P_{F}$ for each $j$.

So, if we also set $S=\oplus S_{p}$ and $P_{S}=\prod S_{p}$, then in the
obvious isomorphism

$Q=P/T\cong P_{F}/F\oplus P_{S}/S,$

each $\phi(y_{j})\in P_{F}/F$ for $j=1,\dots,n$.

It follows that in the corresponding decomposition

${\rm Hom}(D,Q)\cong{\rm Hom}(D,P_{F}/F)\oplus{\rm Hom}(D,P_{S}/S)$

that we can view $\phi$ is an element of the first summand. Therefore, in the
other corresponding decomposition

${\rm Ext}(D,T)\cong{\rm Ext}(D,F)\oplus{\rm Ext}(D,S),$

we can think of $\xi$ also as being an element of the first summand. This
immediately implies the required decomposition, as asserted. ∎

We continue with the following perhaps unsurprising result, a case of which
will be used in the sequel.

######  Proposition 3.2.

Suppose $G$ is a group of finite torsion-free rank with bounded $p$-torsion.
If $K$ is any subgroup of $G$ such that $G/K$ is torsion, then $G/K$ is
actually totally projective. In fact, for each prime $p$, $(G/K)_{p}\cong
G_{p}/K_{p}$ will be the direct sum of a divisible group and a bounded group.

###### Proof.

Let $\overline{G}=G/K$ and $\pi:G\to\overline{G}$ be the canonical
epimorphism. There is clearly no loss of generality in assuming that $G$ and
$K$ are $p$-local groups. It follows that there is a direct sum decomposition
$G=T\oplus F$, where $F$ is torsion-free (and $p$-local). It readily follows
from the theory of finite rank torsion-free $p$-local groups that $\pi(F)\cong
F/[F\cap K]$ is the direct sum of a divisible module and a bounded module
(this follows since $F/(A\cap F)$ must have finite $p$-rank). It is also
obvious that $\pi(T)$ is bounded. An elementary argument then shows that
$\overline{G}=\pi(F)+\pi(T)$ is also the direct sum of a divisible module and
a bounded module. ∎

We now restrict our attention to a particular subclass of the groups described
in Theorem 3.1. Recall once again that the group $G$ is B+E if $G=B\oplus E$,
where $B$ is Bassian and $E$ is elementary. The following gives an alternate
characterization of these groups.

######  Proposition 3.3.

A group $G$ is B+E if, and only if, it has finite torsion-free rank and
$pT_{p}$ is finite for every prime $p$.

###### Proof.

Certainly, the above two conditions are necessary for $G$ to be a B+E-group,
so suppose $r_{0}(G)$ is finite and $pT_{p}$ is finite for all prime $p$. It
follows that the hypotheses of Theorem 3.1 hold; so suppose $G=H\oplus S$ as
in that result. Note that each $S_{p}$ has only a finite number of summands of
the form $\mathbb{Z}_{p^{k}}$ for $k>1$. Transferring each of these from $S$
to $H$ gives the required decomposition into a Bassian group and an elementary
group. ∎

As a direct consequence, we manifestly obtain that the class of B+E-groups is
closed under taking subgroups.

######  Corollary 3.4.

Every subgroup of a B+E-group is also a B+E-group.

###### Proof.

The two conditions in Proposition 3.3 are clearly inherited by all subgroups.
∎

Again, the difference between the B+E-groups and those described in Theorem
3.1 is that, in the later result, we have that each $T_{p}$ is bounded, but if
$G$ is B+E, we demand that each $pT_{p}$ is finite.

We know that if $G$ is generalized Bassian, then it has bounded $p$-torsion;
in fact, for every prime $p$, $pT_{p}$ must be finite. The following
statement, which is a positive answer to a question from [3], shows that such
a group must also have finite torsion-free rank.

######  Theorem 3.5.

A generalized Bassian group $G$ has finite torsion-free rank.

###### Proof.

Suppose $\kappa:=r_{0}(G)$ is infinite. We can clearly find a subgroup
$C\subseteq G$ such that $|C|=\kappa$ and $G=C+T$. We know for each prime $p$
there is a decomposition $T_{p}=R_{p}\oplus S_{p}$, where $R_{p}$ is finite
and $S_{p}$ is elementary. Let $A=C+({\oplus R_{p}})$. Note that $A$ has
cardinality $\kappa$ and

${\oplus R_{p}}\subseteq(T\cap A)\subseteq T=({\oplus R_{p}})\oplus({\oplus
S_{p}}).$

It follows that we can find $E\subseteq{\oplus S_{p}}$ such that $T=(T\cap
A)\oplus E$. Therefore, $A\cap E\subseteq(T\cap A)\cap E=0$ and
$G=C+T\subseteq A+T=A+(T\cap A)+E=A+E\subseteq G$; so we must have $G=A\oplus
E$.

Let $F$ be a free subgroup of $A$ with $A/F$ is torsion. If $D$ is a divisible
hull of $A$, then it easily follows that $r_{0}(F)=r_{0}(D)=|D|=\kappa$ and
there is a surjective homomorphism $F\to D$, which we can extend to a
surjective homomorphism $A\to D$. If $K$ is the kernel of this homomorphism,
then $G/K\cong D\oplus E$. Since there is an obvious embedding

$G=A\oplus E\to D\oplus E\cong G/K$

and the group $G$ is generalized Bassian, there must be a decomposition

$G\cong D\oplus E\oplus K.$

This obviously implies that $G$ has a summand that is a torsion-free divisible
group of infinite rank, which cannot be the case (since if $X$ is a cyclic
subgroup of such a summand $Z$, then $Z$ embeds in $Z/X$, so that $G$ clearly
embeds in $G/X$; but $X$ is not a summand of $G$). ∎

The next consequence immediately follows from a combination of Theorem 3.5 and
Proposition 3.3.

######  Corollary 3.6.

Every generalized Bassian group is B+E.

We now pause for a couple of observations about summands. Suppose $G=A\oplus
B$ and $\gamma:B\to A$ is any homomorphism. If we set

$B^{\prime}=\\{(\gamma(x),x):x\in B\\},$

then it is well known that $G=A\oplus B^{\prime}$; so, in particular, that
$B^{\prime}$ is another summand of $G$. In addition, if $\pi:G=A\oplus B\to B$
is the obvious projection, then suppose $M$ is a subgroup of $G$ such that
$\pi$ restricted to $M$ is injective and there is a decomposition
$B=\pi(M)\oplus K$. Then clearly, $M\cap K=0$ and if $B^{\prime}=M\oplus K$,
then there is a decomposition $G=A\oplus B^{\prime}$. Actually, this second
observation follows from the first: Let $\tau:G\to A$ be the other projection
and define $\gamma:B=\pi(M)\oplus K\to A$ to be 0 on $K$ and
$\tau\circ\pi^{-1}$ on $\pi(M)$.

Preparing for the proof of Theorem 3.8, suppose $G$ is a B+E-group, $N$ is a
subgroup of $G$ and there is an injection $G\to G/N$. Our injection shows
$r_{0}(G)\leq r_{0}(G/N)\leq r_{0}(G)$, which implies that $N$ has no elements
of infinite order. So, $N\subseteq T$ and $T/N$ is the torsion subgroup of
$G/N$.

For each prime $p$,

$p(T/N)_{p}\cong[pT_{p}+N_{p}]/N_{p}\cong pT_{p}/[pT_{p}\cap N_{p}]$

has at most as many elements as $pT_{p}$, which is finite. Therefore, the
derived injection $pT_{p}\to p(T/N)_{p}$ must be an isomorphism. Thus,
$N_{p}\cap pT_{p}=0$, i.e., $N_{p}$ is an elementary summand of $T_{p}$. In
addition, the image of $T_{p}\to G/N$ must be a pure in $(T/N)_{p}$, and hence
a summand. So, we have proved the following technicality.

######  Lemma 3.7.

Suppose $G$ is a B+E-group, $N$ is a subgroup of $G$ and $G\to G/N$ is an
injective homomorphism. Then $N$ is an elementary group that is a summand of
$T$. Further, if $\hat{T}$ is the image of $T$ under this injection, then
$\hat{T}$ is a summand of $T/N$ such that, for all primes $p$,
$p\hat{T}_{p}=p(T/N)_{p}$.

The following result is key to showing that the B+E-groups in many classes
satisfy Conjecture 1.3.

######  Theorem 3.8.

Suppose $G=A\oplus E$ is a B+E-group, where $E$ is elementary and every
elementary summand of $A$ is finite. Then $G$ is generalized Bassian.

###### Proof.

We observe first that $A$ must be Bassian: Certainly, since $G$ has finite
torsion-free rank, so does $A$. Let $p$ be a prime; we need to show $T_{A,p}$
is finite. Since $A$ has no infinite elementary summands, we can conclude that
$f_{A}^{p}(0)$ is finite. And since $p(T_{A,p})\subseteq p(T_{G,p})$, which is
finite, we can conclude that $p(T_{A,p})$ is finite. These facts readily imply
that $T_{A,p}$ is finite, as required.

Suppose $N$ is a subgroup of $G$ such that there is an embedding $G\to G/N$;
let $\hat{G}\subseteq G/N$ be the image of this embedding, $\hat{T}$ be the
torsion subgroup of $\hat{G}$ and $\hat{A}\subseteq\hat{G}$ be the summand of
$\hat{G}$ corresponding to $A\subseteq G$. So, the conclusions of Lemma 3.7
must all hold.

We know that $N$ is a summand of $T$ (we need to show it is a summand of $G$).
Since $N$ is elementary, there is a decomposition $N=N_{A}\oplus N^{\prime}$
where $N_{A}=N\cap A$. Since $A$ is Bassian, for each prime $p$, $N_{A,p}$ is
finite.

On $N^{\prime}$, the projection $G=A\oplus E\to E$ is injective and its image
is a summand of $E$. So there is a decomposition $G=A\oplus E^{\prime}$, where
$N^{\prime}\subseteq E^{\prime}$. Replacing $E$ by $E^{\prime}$ and letting
$N_{E}=N^{\prime}$, we may assume there are decompositions $G=A\oplus E$ and
$N=N_{A}\oplus N_{E}$, where $N_{A}\subseteq A$ and $N_{E}\subseteq E$.

There is a natural decomposition $G/N\cong(A/N_{A})\oplus(E/N_{E})$; let
$\pi:G/N\to E/{N_{E}}$ be the obvious projection. Let $\mathcal{P}^{\prime}$
be the set of primes such that $\pi(\hat{A}[p])\neq 0$. We claim that
$\mathcal{P}^{\prime}$ is actually finite. Otherwise, for each
$p\in\mathcal{P}^{\prime}$, let $x_{p}\in\hat{A}[p]$ with $\pi(x_{p})\neq 0$.
If $M=\oplus_{p\in\mathcal{P}^{\prime}}\langle x_{p}\rangle\subseteq\hat{A}$,
then $\pi$ is an injection on $M$ and $\pi(M)$ is a summand of the elementary
group $E/N_{E}$. Therefore, $M$ will be a summand of $G/N$, and hence of
$\hat{A}$. This, however, contradicts that $A$ has no infinite elementary
summands.

Let $\mathcal{P}$ be the cofinite collection of primes not in
$\mathcal{P}^{\prime}$. It follows that for all $p\in\mathcal{P}$ that
$\hat{A}[p]\subseteq A/{N_{A}}$. So for such $p\in\mathcal{P}$ we have

$f_{A}^{p}(0)=f_{\hat{A}}^{p}(0)\leq
f_{A/{N_{A}}}^{p}(0)=f_{A}^{p}(0)-r_{p}({N_{A}});$

here the first equality is obvious, the next inequality is due to the fact
that the torsion part of $\hat{A}$ is pure in $G/N$ and the last equality is
due to the fact that $N_{A}$ is a summand of $T$. Consequently, for all
$p\in\mathcal{P}$, $(N_{A})_{p}=0$, so that ${N_{A}}$ is finite. Therefore,
${N_{A}}$ is a summand of $A$ and ${N_{E}}$ is a summand of $E$, so that $N$
is a summand of $G$, as required. ∎

We want to generalize Theorem 3.8 by replacing the elementary group $E$ with
any torsion group. Here is the critical step in that program.

######  Proposition 3.9.

Suppose $A$ is a group that has no infinite elementary summands and $S$ is a
torsion group with no elementary summands. Then $A\oplus S$ has no infinite
elementary summands.

###### Proof.

Suppose $E$ is an elementary summand of $A\oplus S$, say $A\oplus S=E\oplus
H$; we must show that $E$ is finite. Let $\pi:A\oplus S\to S$ be the obvious
projection. For any prime $p$, we have

$S[p]=(pS)[p]\subseteq p(E_{p}\oplus H)\subseteq H.$

Therefore, $\pi(E)\subseteq H$. It follows that

$E^{\prime}:=\\{x-\pi(x):x\in E\\}\cong E$

is also a summand of $A\oplus S$. Since $E^{\prime}\subseteq A$, we deduce
that $E^{\prime}$ is also a summand of $A$. So, by hypothesis, $E^{\prime}$
must be finite, which implies that $E$ is finite, as required. ∎

The following is a slight generalization of Theorem 3.8.

######  Corollary 3.10.

Suppose $G$ is a B+E-group with a decomposition $A\oplus S$, where $S$ is
torsion. If $A$ has no infinite elementary summands, then $G$ is generalized-
Bassian.

###### Proof.

As in Theorem 3.8, we can conclude that $A$ is Bassian.

Suppose $S=S_{1}\oplus E$, where $E$ is elementary and $S_{1}$ has no
elementary summand. With Proposition 3.9 at hand, we have $A_{1}=A\oplus
S_{1}$ also has no infinite elementary summand. So, by Theorem 3.8, the group
$G=A_{1}\oplus E$ is generalized Bassian, as expected. ∎

##  4\. Tight subgroups of Bassian groups

The following illustrates how we can use tight subgroups to verify that
certain groups satisfy the hypotheses of Theorem 3.8, so that Conjecture 1.3
holds in this case.

######  Proposition 4.1.

Suppose $A$ is a Bassian group with a tight subgroup $F$. Then any torsion
summand of $A$ is finite.

###### Proof.

Suppose $S$ is such a torsion summand of $A$; since each $S_{p}$ is finite, it
will suffice to show that $S_{p}=0$ for almost all primes $p$. Suppose
$A=A^{\prime}\oplus S$ and $F^{\prime}=F\cap A^{\prime}$; so clearly
$F^{\prime}$ is still free and $A/F^{\prime}$ is torsion. Now, for almost all
primes $p$, the localizations $F_{p}$ and $F_{p}^{\prime}$ will necessarily
coincide. So, for almost all primes $p$, we have

$f_{A^{\prime}}^{p}(j)\geq
f_{F_{p}^{\prime}}^{p}(j)=f_{F_{p}}^{p}(j)=f_{A}^{p}(j)\geq
f_{A^{\prime}}^{p}(j),$

for all $j<\omega$. Therefore, for almost all $p$,
$f_{A}^{p}(j)=f_{A^{\prime}}^{p}(j)$ for all $j$, so that $S_{p}=0$, as
required. ∎

The same proof implies the following consequence.

######  Corollary 4.2.

Suppose $A$ is a Bassian group with a 0-tight subgroup $F$. Then any
elementary summand of $A$ is finite.

######  Corollary 4.3.

If $G$ is a B+E-group with a decomposition $A\oplus S$, where $S$ is torsion
and $A$ has a 0-tight subgroup, then $G$ is generalized-Bassian.

###### Proof.

The existence of a 0-tight subgroup clearly implies that $f_{A}^{p}(0)$ is
finite for all primes. And since each $pA_{p}$ must clearly be finite, we can
conclude that $A$ must be Bassian. By virtue of Corollary 4.2, $A$ has no
infinite elementary summands, so the result follows from Corollary 3.10. ∎

We now observe that our conjecture holds for the class of Warfield groups, and
in particular, the balanced-projective groups.

######  Corollary 4.4.

Suppose $G$ is a Warfield group. Then $G$ is generalized Bassian if, and only
if, it is a B+E-group.

###### Proof.

Follows from a combination of Corollary 4.3 and Proposition 2.2. ∎

For groups of finite torsion-free rank and bounded $p$-torsion, the following
statement describes when they have free quasi-essential subgroups that are
nice.

######  Proposition 4.5.

Suppose $G$ is a group with finite torsion-free rank and bounded $p$-torsion.
Then the following are equivalent:

(a) Every free quasi-essential subgroup $F$ is nice in $G$;

(b) Some free quasi-essential subgroup $F$ is nice in $G$;

(c) For every prime $p$, the quotient $G_{p}/T_{p}$ is the direct sum of a
free local module and a divisible module.

###### Proof.

Suppose that $F$ is an arbitrary free quasi-essential subgroup of $G$. It will
clearly suffice to show that $F$ is nice if, and only if, $G_{p}/T_{p}$, for
all primes $p$, is the direct sum of a free local module and a divisible
module. And since $F$ is nice in $G$ if, and only if, $F_{p}$ is nice in
$G_{p}$ for each prime, it suffice to assume that all groups mentioned are
$p$-local.

Also, there is a splitting $G=T\oplus H$, where $H$ is torsion-free of finite
rank. If $p^{n}T=0$, then since $F(n)\subseteq H$ and $F/F(n)$ is finite, $F$
is nice in $G$ if and only if $F(n)$ is nice in $H$. It follows that there is
no loss of generality in assuming $H=G$ is torsion-free.

If $G=R\oplus D$, where $R$ is reduced and $D$ is divisible, then
$F(\infty)=F\cap D$ is nice in $D$, since every element of a corresponding
coset has height $\infty$. So $F$ is nice in $G$ if and only if
$F_{\infty}:=F/F(\infty)$ is nice in $G/G(\infty)\cong R$. However,
$F(\infty)$ is pure in $F$, and hence a summand, so that $F_{\infty}$ is free.
It therefore follows that $F_{\infty}$ is nice in $R$ if, and only if,
$R/F_{\infty}$ is finite, i.e., if, and only if, $R$ is free, as required. ∎

The following assertion allows us to replace a decomposition subgroup of a
Warfield group with a general free nice quasi-essential subgroup $F\subseteq
G$.

######  Lemma 4.6.

Suppose $F$ is a free finite rank valuated group. Then $F$ has a nice quasi-
essential realization $A$ that is a Bassian group if, and only if, for every
prime $p$, there is an $n_{p}\in\omega$ such that $F_{p}(n_{p})$ has a
valuated decomposition $X_{p}\oplus F_{p}(\infty)$, where for every $x\in
X_{p}\subseteq F_{p}(n_{p})$, $|x|_{p}$ is $n_{p}$ plus the $p$-height of $x$
as an element of $X_{p}$. In fact, if $F$ satisfies this condition for each
prime $p$, then we can construct $A$ containing $F$ as a 0-tight subgroup.

###### Proof.

If we are given the nice realization $A$, one argues as in Proposition 4.5
that each $F_{p}$ has the required decomposition (or see the proof of Theorem
4.7). So, assume we are given $F$ so that such decompositions always exist.
There is clearly no loss of generality in assuming that for each prime $p$,
$n_{p}\geq 2$.

For each such prime, let $b_{p,\infty,j}\in F$, for $j=1,\dots,d_{p,\infty}$,
be a $\mathbb{Z}_{(p)}$-basis for the free module $F_{p}(\infty)$. Define a
set of generators $y_{p,\infty,j,i}$ ($i<\omega$) subject to the relations
$y_{p,\infty,j,0}=b_{p,\infty,j}$ and $py_{p,\infty,j,i+1}=y_{p,\infty,j,i}$.

Similarly, let $b_{p,n_{p},j}\in F$, for $j=1,\dots,d_{p,n_{p}}$, be a
$\mathbb{Z}_{(p)}$-basis for $X_{p}$. Define a set of generators
$y_{p,n_{p},j}$ subject to the relations
$p^{n_{p}}y_{p,n_{p},j}=b_{p,n_{p},j}$.

Next, for $m=2,\dots n_{p}-1$, let $b_{p,m,j}\in F$, for $j=1,\dots,d_{p,m}$,
project onto a basis for $F_{p}(m)/F_{p}(m+1)$. Define a set of generators
$y_{p,m,j}$ subject to the relations $p^{m}y_{p,m,j}=b_{p,m,j}$.

Finally, let $b_{p,1,j}\in F$, for $j=1,\dots,d_{p,1}$, project onto a basis
for

$F_{p}(1)/(pF_{p}+F_{p}(2)).$

Define a set of generators $y_{p,1,j}$ subject to the relations
$py_{p,1,j}=b_{p,1,j}$.

We let $A$ be the group generated by $F$ together with all the $y$s subject to
the above relations. Note that for all primes $p$, $A_{p}$ is the $p$-local
module generated by $F_{p}$ and all of the above generators of the form
$y_{p,\dots}$. We want to show that $A$ is a realization of $F$, i.e., for all
primes $p$ and all $\alpha\in\omega\cup\\{\infty\\}$, we have
$F_{p}(\alpha)=p^{\alpha}A_{p}\cap F_{p}$.

If $U_{p}$ is the submodule of $A_{p}$ generated by $F_{p}(\infty)$ together
with $y_{p,\infty,j,i}$ for all $j\leq d_{p,\infty}$, $i<\omega$, then it is
easy to see that $U_{p}$ is divisible, torsion-free and
$F_{p}(\infty)=F_{p}\cap U_{p}$. In fact, $U_{p}$ will be a divisible hull of
$F_{p}(\infty)$ and hence a summand of any containing module.

Let $W\subseteq A$ be the subgroup generated by $F$ and all possible
$y_{p,\infty,j,i}$. Setting $|y_{p,\infty,j,i}|_{p}=\infty$ and
$|y_{p,\infty,j,i}|_{q}=|b_{p,\infty,j}|_{q}$ for all $j\leq d_{p,\infty}$,
$i<\omega$ and $q\neq p$ extends the valuation on $F$ to one on $W$ with
$W_{p}(\infty)=p^{\infty}W_{p}=U_{p}$. There is clearly a valuated
decomposition $W_{p}(n_{p})=X_{p}\oplus U_{p}$. It will suffice to show that
$A$ is a realization of $W$; i.e., each $A_{p}$ is a realization of $W_{p}$.

Note that for $m\leq n_{p}$, and $j\leq d_{p,m}$, if $p^{s}y_{p,m,j}\in
W_{p}$, then $s\geq m$ and $|p^{s}y_{p,m,j}|_{W_{p}}\geq s$. It follows that
$p^{s}A_{p}\cap W_{p}\subseteq W_{p}(s)$ for all $s<\omega$.

We show by a reverse induction that $W_{p}(s)\subseteq p^{s}A_{p}\cap W_{p}$;
i.e., $W_{p}(s)\subseteq p^{s}A_{p}$. We first verify this for $s=n_{p}$. In
the decomposition $W_{p}(n_{p})=X_{p}\oplus U_{p}$, since the $b_{p,n_{p},j}$
map on to a $\mathbb{Z}_{(p)}$-basis for $X_{p}$ it follows that

$W_{p}(n_{p})=\langle b_{p,n_{p},j}\rangle+U_{p}=\langle
p^{n_{p}}y_{p,n_{p},j}\rangle+p^{n_{p}}U_{p}\subseteq p^{n_{p}}A.$

So, if $s=n_{p}+k\geq n_{p}$, then $W_{p}(s)=p^{k}W_{p}(n_{p})\subseteq
p^{k}p^{n_{k}}A=p^{s}A,$ as required.

Suppose now that $1<s<n_{p}$ and this holds for $s+1$. Then

$W_{p}(s)=\langle b_{p,s,j}\rangle+W_{p}(s+1)\subseteq
p^{s}A_{p}+p^{s+1}A_{p}=p^{s}A_{p}.$

Finally, if $s=1$ and it has been verified for $s=2$, then

$W_{p}(1)=\langle b_{p,1,j}\rangle+pW_{p}+W_{p}(2)\subseteq
pA_{p}+pW_{p}+p^{2}A_{p}=pA_{p}.$

Therefore, $A$ is a realization of $W$ and $F$.

Note that the quotient $A/W$ is a torsion group. If $T$ is the torsion
subgroup of $A$ and $p$ is a prime, then $T_{p}\cap W_{p}=0$. So $T_{p}$
embeds in $A_{p}/W_{p}$ which is clearly finite. Therefore, each $T_{p}$ is
finite, so that $A$ is Bassian.

We next verify that $V$ is nice in $A$. To that end, let $p$ some prime.
Clearly, $V_{p}$ is nice in $W_{p}$, since every coset has an element of
infinite value. And $W_{p}$ is nice in $A_{p}$, since the corresponding
quotient is finite. Therefore, each $V_{p}$ is nice in $A_{p}$, as required.

We now claim that $F$ is $0$-tight in $A$. To that end, suppose $p$ is a prime
and $x\in A$ represents an element of $U_{0}(A)$; i.e., $|x|_{A}=0$ and
$|px|_{A}\geq 2$; we need to show that $x\in F+pA$.

Since $F$ and $F_{p}$, as well as $A$ and $A_{p}$ have isomorphic Ulm
invariants, there is no loss of generality in assuming $F$ and $A$ are local.
So, we may assume that $x$ is the sum of an element of $F$ plus terms of the
form $sy_{p,m,j}$ and $ty_{p,\infty,j,i}$ where such $s,t$ are integers. Since
each $y_{p,\infty,j,i}\in pA$, modulo $pA$, we may again ignore the last type
of term. Similarly, whenever $m>1$, since $|px|_{A}\geq 2$, we can conclude
$p$ divides $s$, so again modulo $pA$ we may ignore terms of the form
$sy_{p,m,j}$ for $m>1$, as well. So we may assume $x$ is of form

$x=s_{1}y_{p,1,1}+\cdots+s_{n_{1}}y_{p,1,n_{1}}+v\ \ \ (v\in F).$

Since

$px=s_{1}b_{p,1,1}+\cdots+s_{n_{1}}b_{p,1,d_{p,1}}+pv\in F$

has height at least $2$, $px-pv=s_{1}b_{p,1,1}+\cdots
s_{n_{1}}b_{p,1,d_{p,1}}$ represents the zero element of
$F_{p}(1)/(pF_{p}+F_{p}(2))$. Since these $b$s were chosen to project to a
basis of this vector space, we can conclude that $p$ divides each $s$.
Therefore, each $s_{i}y_{p,1,i}$ is in $\langle b_{p,1,i}\rangle\subseteq F$,
so that $x\in F$, as required. ∎

The following result clearly generalizes the fact that any Warfield B+E-group
is necessarily generalized Bassian. It allows us to replace a decomposition
subgroup by any nice free quasi-essential subgroup.

######  Theorem 4.7.

Suppose $G$ is a group with a free nice quasi-essential subgroup $F$. Then $G$
is generalized Bassian if, and only if, it is a B+E-group.

###### Proof.

One direction being immediate, suppose $G$ is a B+E-group. Note that by
Proposition 3.2, $G/F$ is totally projective, i.e., for each prime
$(G/F)_{p}\cong G_{p}/F_{p}$ is the direct sum of a divisible and a bounded
module.

We claim that $F$ (with the induced valuations) satisfies Lemma 4.6, so assume
$p$ is some prime; we need to find $n_{p}$ and a decomposition
$F_{p}=X_{p}\oplus F_{p}(\infty)$ as in that result.

Owing to Proposition 4.5, there is a decomposition

$G_{p}=T_{p}\oplus R_{p}\oplus D,$

where $R_{p}$ is a free module and $D$ is divisible. If $p^{m}T_{p}=0$, then
$F_{p}(m)\subseteq R_{p}\oplus D$. Find $n_{p}\geq m$ such that
$X_{p}:=p^{n_{p}}R_{p}\subseteq F_{p}$. Since

$p^{n_{p}}(R_{p}\oplus D)=(p^{n_{p}}R_{p})\oplus D,$

we can conclude that $F_{p}(n_{p})=X_{p}\oplus F_{p}(\infty)$, and our
decomposition behaves as required.

So, in view of Lemma 4.6, we can find a Bassian group $A$ that is a nice
$0$-tight realization of $F$. Let $T^{1}$ be a second copy of the torsion
subgroup of $A$ and $T^{2}$ be a second copy of the torsion subgroup of $G$.
Since the Ulm invariants of $G\oplus T^{1}$ agree with those of $A\oplus
T^{2}$, and the Ulm invariants of the valuated group $F$ are all finite, it
follows that the relative Ulm invariants of $F$ in $G\oplus T^{1}$ agree with
those of $F$ in $A\oplus T^{2}$. [Notice that, for a given prime $p$, both
relative Ulm invariants will equal $f^{p}_{T^{1}\oplus
T^{2}}(\alpha)-f^{p}_{F}(\alpha)$ when $\alpha$ is finite and $0$ otherwise.]

Therefore, the identity isometry $F\cong F$ extends to an isomorphism $G\oplus
T^{1}\cong A\oplus T^{2}$. It follows from Corollary 4.3 that $A\oplus T^{2}$
is generalized Bassian. Consequently, $G$, as a summand of a generalized
Bassian group, will also be a generalized Bassian group, as required. ∎

As a valuable consequence, we derive the following.

######  Corollary 4.8.

Suppose $G$ is a group such that $G/T$ is divisible. Then $G$ is generalized
Bassian if, and only if, it is a B+E-group.

###### Proof.

Again, the necessity is clear. On the other hand, if $G$ is B+E, then
according to Proposition 4.5, it has a free nice quasi-essential subgroup. So,
the result follows at once from Theorem 4.7, as desired. ∎

Of the groups mentioned in Section 2, we have already noted that Conjecture
1.3 holds for Warfield (and balanced-projective) group. It follows directly
from Corollary 4.8 that it also holds for the other class mentioned there.

######  Corollary 4.9.

Suppose $G$ is a PSP-group. Then $G$ is generalized Bassian if, and only if,
it is a B+E-group.

We have already mentioned that the class of B+E-groups is closed under
arbitrary subgroups. The following makes this more explicit and demonstrates
an even closer connection between those groups and the generalized Bassian
groups.

######  Corollary 4.10.

A group $G$ is B+E if, and only if, it embeds in a group that is generalized
Bassian.

###### Proof.

We know that any generalized Bassian group is B+E and that a subgroup of a
B+E-group retains that property, so that sufficiency is clear.

Conversely, if $G$ is a B+E-group, and $D$ is a (torsion-free finite rank)
divisible hull of $G/T$, then the surjectivity of the map

${\rm Ext}(D,T)\to{\rm Ext}(G/T,T)$

implies that there is a group $G^{\prime}$ containing $G$ (and hence $T$) such
that $G^{\prime}/T\cong D$ is divisible. Clearly, $G^{\prime}$ will be a
B+E-group. It follows from Corollary 4.8 that $G^{\prime}$ is generalized
Bassian, concluding the proof. ∎

Let us recall now that a group $G$ is Hopfian if, for any (possibly proper)
subgroup $H$ of $G$, whenever $G\cong G/H$, it must be that $H=\\{0\\}$. It is
principally known that all Bassian groups are Hopfian (see, e.g., [2]), which
is not the case for generalized bassian groups, however (see, e.g., [3]).
Nevertheless, the following necessary and sufficient condition is true, which
proof is not so hard and follows from our results alluded to above, so we will
omit the details in its proving.

######  Proposition 4.11.

A B+E-group is Bassian if, and only if, it is Hopfian. In particular, the same
holds for generalized Bassian groups.

Funding: The work of the first-named author, P.V. Danchev, is partially
supported by the Bulgarian National Science Fund under Grant KP-06 No. 32/1 of
December 07, 2019 as well as by the Junta de Andalucía, Grant FQM 264, and by
the BIDEB 2221 of TÜBÍTAK.

## References

  * [1] A. Bass, A finiteness property of affine algebras, Proc. Amer. Math. Soc. 110(2) (1990), 315–318.
  * [2] A.R. Chekhlov, P.V. Danchev and B. Goldsmith, On the Bassian property for Abelian groups, Arch. Math. (Basel) 117(6) (2021), 593–600.
  * [3] A.R. Chekhlov, P.V. Danchev and B. Goldsmith, On the generalized Bassian property for Abelian groups, Acta Math. Hungar. 168(1) (2022), 186–201.
  * [4] L. Fuchs, Abelian Groups, Springer, Switzerland (2015).
  * [5] P. Hill and C. Megibben, Mixed groups, Trans. Amer. Math. Soc. 334 (1992), 121–142.
  * [6] R. Hunter and F. Richman, Global Warfield groups, Trans. Amer. Math. Soc. 266(1) (1981), 555–572.
  * [7] R. Hunter and E. Walker, S-groups revisited, Proc. Amer. Math. Soc. 82(1) (1981), 13–18.
  * [8] P. Loth, Charactetizations of Warfield groups, J. Algebra 204 (1998), 32–41.
  * [9] F. Richman and E.A. Walker, Valuated groups, J. Algebra 56(1) (1979), 145–167.
  * [10] R. B. Warfield, Jr., Classification theory of abelian groups. I. Balanced projectives, Trans. Amer. Math. Soc. 222 (1976), 33–63.


∎

11institutetext:  aInstitute of Physics, University of Brasília, P.O. Box
04531, 70.919-970, Brasília, DF, Brazil.
bDepartment of Electrical Engineering, University of Brasília, 70.919-970,
Brasília, DF, Brazil.
Corresponding Author: 11email<EMAIL_ADDRESS>
# Evaluation of Peppermint Leaf Flavonoids as SARS-CoV-2 Spike Receptor-
Binding Domain Attachment Inhibitors to the Human ACE2 Receptor: A Molecular
Docking Study

M. L. Pereira Júniora, R. T. de Sousa Juniorb G. D. Amvame Nzeb W. F. Giozzab
L. A. Ribeiro Júniora

(Received: xx/xx/2021 / Accepted: xx/xx/2021)

###### Abstract

Virtual screening is a computational technique widely used for identifying
small molecules which are most likely to bind to a protein target. Here, we
performed a molecular docking study to propose potential candidates to prevent
the RBD/ACE2 attachment. These candidates are sixteen different flavonoids
present in the peppermint leaf. Results showed that Luteolin
7-O-neohesperidoside is the peppermint flavonoid with a higher binding
affinity regarding the RBD/ACE2 complex (about -9.18 Kcal/mol). On the other
hand, Sakuranetin presented the lowest affinity (about -6.38 Kcal/mol).
Binding affinities of the other peppermint flavonoids ranged from -6.44
Kcal/mol up to -9.05 Kcal/mol. The binding site surface analysis showed
pocket-like regions on the RBD/ACE2 complex that yield several interactions
(mostly hydrogen bonds) between the flavonoid and the amino acid residues of
the proteins. This study can open channels for the understanding of the roles
of flavonoids against COVID-19 infection.

Keywords
Coronavirus, Sars-CoV-2, Peppermint Flavonoids, RBD/ACE2 Inhibitors.

##  1 Introduction

The COVID-19 is an infectious disease caused by the coronavirus SARS-CoV-2
chen_TL ; rothan_JA ; guo_MMR ; tu_IJMS ; wang_IJAA . It has reached the
status of a pandemic in March of 2020. Up to January of 2021, it has already
infected more than 100 million people, leading to the death of more than 2
million ones oms_url . Since the earlier stages of this pandemic, a worldwide
effort has been devoted to producing vaccines and antiviral drugs to combat
this virus. Some successful investigations yielded vaccines that have started
to be applied very recently sharma_FiI ; voysey_TheLancet ; corbett_NEJM ;
baden_NEJM ; anderson_NEJM ; zhang_medrxiv ; jones_TheLancet ; burki_TheLancet
; jones_TheLacent . Despite the beginning of vaccination, no consensus about
an efficient treatment for already infected patients has been reached so far.

Sars-CoV-2 has a crown-like (spherical) form, and its surface protein (Spike)
is directly involved in the infectious process wu_BBaI ; chhikara_CBL ;
liang_FiI . The receptor of this virus in human cells is the angiotensin-
converting enzyme 2 (ACE2) zhang2020angiotensin ; li2003angiotensin ;
kuba2005crucial . Sars-CoV-2 surface protein has two subdivisions named S1 and
S2, being S1 the receptor-binding domain (RBD) xiu2020inhibitors ;
tai2020characterization ; spinello2020rigidity ; han2020computational . The
RBD plays a major role in the attachment mechanism of Spike protein to ACE2
singh2020serine . After the attachment between them, the virus enters the cell
and starts the replication process xiu2020inhibitors . In this sense, the
strategy of virtual screening for possible inhibitors for the RBD/ACE2
attachment basu2020molecular  may pave the way for novel therapeutic
approaches for the treatment of COVID-19.

Drug repurposing is a feasible way to combat diseases with some similarities
zhou2020network ; gordon2020sars ; pandey2020potential . In this scenario, the
use of phytochemicals is always an important option to be considered
swain2020phytochemicals . Among their sub-classes, the flavonoids — a class of
small molecules found in fruits, vegetables, flowers, honey, teas, and wines —
stand out panche2016flavonoids ; harborne2013flavonoids ; hertog1992content .
Their pharmacological properties include antimicrobial, antioxidant, anti-
inflammatory, and antiviral functions kumar2013chemistry ;
sharma2006pharmacological ; kim2004anti .

Flavonoids have been employed as inhibitors for the infection mechanism of
several diseases cushnie2005antimicrobial . Among them, one can mention
malaria, leishmaniasis, Chagas, and dengue lehane2008common ; muzitano2009oral
; marin2011vitro ; sanchez2000antiviral ; kiat2006inhibitory ;
de2015flavonoids . They have also been considered in studies aimed at
developing therapeutic approaches for cancer treatment abotaleb2019flavonoids
; wang2000therapeutic ; eunjung2012 . Very recently, it was reported that
Luteolin (a flavonoid found in leaves and shells) is efficient as an anti-
inflammatory that can interact with the Sars-CoV-2 surface yu2020computational
and its main protease muchtaridi2020natural . More specifically, it is
adsorbed in the Spike protein, inhibiting the Sars-CoV-2 attachment to the
ACE2, thus preventing infection. Ngwa and colleagues used computer simulations
to address the feasibility of Caflanone, Hesperetin, and Myricetin flavonoids
in acting as inhibitors for the ACE2 active site attachment ngwa_M . Their
results pointed to the ability of Caflanone in inhibiting the transmission of
the Sars-CoV-2 virus from mother to fetus in pregnancy. Pandey et. al.
conducted molecular docking and dynamics simulations considering ten flavonoid
and non-flavonoid compounds (by using phytochemicals and hydroxychloroquine,
respectively) to verify their performance in inhibiting the RBD/ACE2
interaction pandey_JBSD . Their findings indicate that Fisetin, Quercetin, and
Kamferol molecules couple to RBD/ACE2 complex with good binding affinities. In
this sense, they can be explored as possible anti-Sars-CoV-2 agents. Despite
the success of these molecules inhibiting the RBD/ACE2, other flavonoids
should be tested to broaden the list of possible inhibitors and to confirm
their potential in developing new therapeutic approaches for the treatment of
COVID-19.

Herein, in silico molecular docking analysis was carried out to propose
potential flavonoid candidates in preventing the RBD/ACE2 attachment. These
candidates are sixteen different flavonoids present in the Peppermint (Mentha
piperita) leaf DOLZHENKO201067 ; McKayAreviewof ; bodalska2020analysis ;
areias2001phenolic ; riachi2015peppermint ; mahendran2020ethnomedicinal ;
peterson1998flavonoids . Peppermint is a perennial herb and medicinal plant
native to Europe widely used for treating stomach pains, headaches, and
inflammation of muscles McKayAreviewof ; mahendran2020ethnomedicinal ;
peterson1998flavonoids . Well-known for their flavoring and fragrance traits,
peppermint leaves and the essential oil extracted from them are used in food,
cosmetic and pharmaceutical products McKayAreviewof ; bodalska2020analysis ;
DOLZHENKO201067 ; areias2001phenolic . Our results revealed that Luteolin
7-O-neohesperidoside is the peppermint flavonoid with a higher binding
affinity regarding the RBD/ACE2 complex (about -9.18 Kcal/mol). On the other
hand, Sakuranetin was the one with the lowest affinity (about -6.38 Kcal/mol).
Binding affinities of the other peppermint flavonoids ranged from -6.44
Kcal/mol up to -9.05 Kcal/mol. These binding affinities are equivalent to
other ones reported in the literature for the interaction between flavonoids
and the RBD/ACE2 complex DeOliveira2020 ; Muhseen2020 ; Hu2020 ;
muchtaridi2020natural ; Basu2020 ; Istifli2020 ; Peterson2020 ;
yu2020computational ; dourav_JBSD ; RUSSO2020109211 . Moreover, the binding
site surface analysis showed pocket-like regions on the RBD/ACE2 complex that
yield several interactions (mostly hydrogen bonds) between the flavonoid and
the amino acid residues of the proteins. Definitively, experimental studies
and clinical trials should be further performed to evaluate the efficacy of
these compounds in the inhibition of the RBD/ACE2 attachment.

##  2 Materials and Methods

Figure 1: Schematic representation of the (a) main proteins involved RBD/ACE2
interaction. These proteins were obtained from Protein Data Bank, ID 6M0J
lan2020structure . (b) The binding site surface has the following color
scheme: gray, red, blue, and white for carbon, oxygen, nitrogen, and hydrogen
atoms, respectively. Only the three regions (R1, R2, and R3) were considered
in the docking processes since they define the whole RBD/ACE2 interface. The
TYR4, GLN24, PHE486, and ASN487 are the residues present in the region R1;
GLU35, LYS31, and GLN493 are the residues present in the region R2; ASP355,
THR500, GLY502, GLN498, LYS353, and ASP38 are the residues present in the
region R3.

Since Sars-CoV-2 infects human cells through the RBD/ACE2 coupling, the idea
of checking for small molecules that may inhibit this interaction is recurring
and can be useful to propose a combatant drug Bojadzic2020.10.22.351056 .
Here, we used molecular docking to study the interaction between the
peppermint flavonoids with the RBD/ACE2 complex. Below, we present the
proteins, inhibitors (flavonoids), and the computational protocol involved in
our study.

###  2.1 Protein Preparation

Figure 1 presents the main proteins involved RBD/ACE2 interaction that were
obtained from Protein Data Bank, ID 6M0J lan2020structure . In the left panel
of this figure, the ACE2 protein is in blue, while the RBD Sars-CoV-2 one is
in red. Three essential regions of inhibition between these proteins were
highlighted with the black squares R1, R2, and R3. In the right side of Figure
1 we show the binding site surface colored as gray, red, blue, and white for
carbon, oxygen, nitrogen, and hydrogen atoms, respectively. The yellow
rectangle highlights the total surface for inhibition with a clear cavity
within region R2. The protein resolution is 2.45 Å, and no pKa prediction was
carried out. The modeled structure has 41 residues less than the deposited
one, but all the important residues in the RDB/ACE2 interface were considered
in our study. Just metal ions were considered in the docking study, water
molecules were not included.

###  2.2 Ligand Preparation

Figure 2: Chemical structure of peppermint leaf flavonoids: (a )Acacetin, (b)
Apigenin, (c) Apigenin 7-O-neohesperidoside (Apigenin*), (d) Chryseoriol, (e)
Hesperidin, (f) Hesperitin, (g) Ladanein, (h) Luteolin, (i) Luteolin
7-O-glucoside (Luteolin*), (j) Luteolin 7-O-glucuronide (Luteolin**), (k)
Luteolin 7-O-neohesperidoside (Luteolin***), (l) Narigenin, (m) Pebrellin, (n)
Sakuranetin, (o) Thymusin, and (p) Xanthomicrol.

The peppermint leaf contains sixteen flavonoids DOLZHENKO201067 ;
areias2001phenolic , classified into three subcategories: Flavones
(Flavonols), Flavorings, and Flavanones DOLZHENKO201067 ; areias2001phenolic .
The flavonoids studied here are Acacetin, Apigenin, Apigenin
7-O-neohesperidoside (Apigenin*), Chryseoriol, Hesperidin, Hesperitin,
Ladanein, Luteolin, Luteolin 7-O-glucoside (Luteolin*), Luteolin
7-O-glucuronide (Luteolin**), Luteolin 7-O-neohesperidoside (Luteolin***),
Narigenin, Pebrellin, Sakuranetin, Thymusin, and Xanthomicrol. Their 3D
structures were extracted from PubChem pub_url . The chemical structures of
these flavonoids can be seen in figure 2, while relevant information such as
PubChem ID, molecular weight, molecular formula, and subcategory of the
flavonoid is presented in table 1.

Compound | PubChem CID | Mol. Weight (g/mol) | Mol. Formula | Type
---|---|---|---|---
Acacetin | 5280442 | 284.26 | C16H12O5 | Flavones and Flavonols
Apigenin | 5280443 | 270.24 | C15H10O5 | Flavones and Flavonols
Apigenin* | 5282150 | 578.5 | C27H30O14 | Flavones and Flavonols
Chryseoriol | 5280666 | 300.26 | C16H12O6 | Flavones and Flavonols
Hesperidin | 10621 | 610.6 | C28H34O15 | Flavorings
Hesperitin | 72281 | 302.28 | C16H14O6 | Flavanones
Ladanein | 3084066 | 314.29 | C17H14O6 | Flavones and Flavonols
Luteolin | 5280445 | 286.24 | C15H10O6 | Flavones and Flavonols
Luteolin* | 5280637 | 448.4 | C21H20O11 | Flavones and Flavonols
Luteolin** | 5280601 | 462.4 | C21H18O12 | Flavones and Flavonols
Luteolin*** | 5282152 | 594.5 | C27H30O15 | Flavones and Flavonols
Naringenin | 932 | 272.25 | C15H12O5 | Flavorings
Pebrellin | 632255 | 374.3 | C19H18O8 | Flavones and Flavonols
Sakuranetin | 73571 | 286.28 | C16H14O5 | Flavanones
Thymusin | 628895 | 330.29 | C17H14O7 | Flavones and Flavonols
Xanthomicrol | 73207 | 344.3 | C18H16O7 | Flavones and Flavonols
Table 1: Potential inhibitors (peppermint leaf flavonoids) of RBD/ACE2 complex
and their compound information.

###  2.3 Molecular Docking Simulation

Molecular docking consists of computationally analyze the non-covalent binding
between macromolecules (receptor) and small molecules (ligand). Here, the
macromolecule is the RBD/ACE2 protein complex (Figure 1), while the ligands
are the sixteen flavonoids present in the peppermint leaf (Figure 2).
SWISSDOCK server was used for the docking simulations grosdidier2011fast ;
grosdidier2011swissdock . In SWISSDOCK, the docking energies are obtained
through the CHARMM (Chemistry at HARvard Macromolecular Mechanics) force field
grosdidier2011swissdock ; grosdidier2011fast  using a blind docking strategy
that spans over 100 trial configurations for each target/ligand input
cheng2020discovery . The target/ligand configuration with higher binding
affinity is selected using the UCFS CHIMERA software pettersen2004ucsf , a
visualization tool capable of directly import data from the SWISSDOCK server.
Finally, the Protein-Ligand Interaction Profiler (PLIP) server
salentin2015plip  is used to characterize the target/ligand interaction for
the configuration with a higher binding affinity for each flavonoid regarding
the RBD/ACE2 complex. It is worth mentioning that the screening for the ligand
position was limited just to the ACE2/RDB interface (regions R1, R2, and R3 in
the left panel of Figure 1). This interface is the crucial region to be
considered for blocking the coronavirus entry and replication cycle. The
simulation (docking) box used in the screening for the ligand position was
limited just to the ACE2/RDB interface (regions R1, R2, and R3 in the left
panel of Figure 1). The docking box has 27.5 Å $\times$ 9.0 Å $\times$ 8.5 Å
of dimension and it was centered at (31.5,-36.0,1.5) Å. These parameters cover
the three regions depicted in Figure 1. The accuracy in estimating the ligand
positions and related binding affinities are $\pm$2 Å and $\pm$0.01 Kcal/mol,
respectively.

##  3 Results

Compound |  $\Delta G$ [Kcal/mol]
---|---
Acacetin | -6.70
Apigenin | -6.87
Apigenin 7-O-neohesperidoside | -8.08
Chryseoriol | -6.78
Hesperidin | -8.67
Hesperitin | -6.80
Ladanein | -6.56
Luteolin | -7.24
Luteolin 7-O-glucoside | -8.01
Luteolin 7-O-glucuronide | -7.74
Luteolin 7-O-neohesperidoside | -9.18
Naringenin | -6.44
Pebrellin | -7.07
Sakuranetin | -6.38
Thymusin | -6.94
Xanthomicrol | -6.83
Table 2: Peppermint leaf-based flavonoid candidates undergoing docking
experiment with their most favorable conformation (lowest binding affinity
$\Delta G$ in Kcal/mol). Figure 3: binding site surface (BSS) for the putative
best docking target/ligand configurations of (a) Acacetin, (b) Apigenin, (c)
Apigenin∗, (d) Chryseoriol, (e) Hesperidin, (f) Hesperetin, (d) Ladanein, and
(d) Luteolin. Figure 4: binding site surface (BSS) for the putative best
docking target/ligand configurations of (a) Luteloin∗, (b) Luteloin∗∗, (c)
Luteloin∗∗∗, (d) Naringenin, (e) Pebrellin, (f) Sakuranetin, (d) Thymusin, and
(d) Xanthomicrol. Figure 5: PLIP docked poses for the RBD/ACE2 interaction
with (a) Acacetin, (b) Apigenin, (c) Apigenin∗, (d) Chryseoriol, (e)
Hesperidin, (f) Hesperetin, (d) Ladanein, and (d) Luteolin. The hydrogen,
hydrophobic, and $\pi$-staking bonds are denoted by the blue, dashed gray, and
dashed yellow lines, respectively. The yellow sphere represents the charge
center. Figure 6: PLIP docked poses for the RBD/ACE2 interaction with (a)
Luteloin∗, (b) Luteloin∗∗, (c) Luteloin∗∗∗, (d) Naringenin, (e) Pebrellin, (f)
Sakuranetin, (d) Thymusin, and (d) Xanthomicrol. The hydrogen, hydrophobic,
and $\pi$-staking bonds are denoted by the blue, dashed gray, and dashed
yellow lines, respectively. The yellow sphere represents the charge center.
ACE2 and RBD moieties are shown above and below the horizontal line,
respectively.

After successful docking of the peppermint flavonoids to the RBD/ACE2 complex,
several modes of ligand/target interactions were generated with a particular
docking score (binding affinity). The binding mode with the lowest binding
affinity is regarded as the best one, once it tends to be the most stable. The
binding affinity results ($\Delta G$) obtained here are summarized in Table 2.
SWISSDOCK simulations for all the ligands in Figure 2 revealed significant
binding affinities with the target RBD/ACE2 proteins. Luteolin
7-O-neohesperidoside is the peppermint flavonoid with a higher binding
affinity regarding the RBD/ACE2 complex (about -9.18 Kcal/mol). On the other
hand, Sakuranetin was the one with the lowest affinity (approximately -6.38
Kcal/mol). Binding affinities of the other peppermint flavonoids ranged from
-6.44 Kcal/mol up to -9.05 Kcal/mol. As one can note in Tables 1 and 2, the
best docked flavonoids have greater molecular weight. All the binding
affinities are close to the ones reported for the RBD/ACE2 interaction with
other species of flavonoids DeOliveira2020 ; Muhseen2020 ; Hu2020 ;
muchtaridi2020natural ; Basu2020 ; Istifli2020 ; Peterson2020 ;
yu2020computational ; dourav_JBSD ; RUSSO2020109211 . Moreover, they can
outperform the binding affinities reported by docking studies using other
types of compounds targeting RBD/ACE2 Basu2020 ; luan2020targeting ;
alexpandi2020quinolines ; choudhary2020identification ; trezza2020integrated ;
wei2020silico ; panda2020structure ; utomo2020revealing , such as Chloroquine
and Hydroxychloroquine, which are lower than -8.0 Kcal/mol Basu2020 . This
fact can be attributed to the abundant phenolic hydroxyl group in flavonoids.
The hydroxyl group in the sugar group of flavonoids tends to bind more easily
with the heteroatoms of amino acids from RBD/ACE2, as will be shown later. In
this sense, peppermint flavonoids can compose the list of potential
phytochemical inhibitors for the RBD/ACE2 interaction.

Figures 3 and 4 illustrate the binding site surface (BSS) for the putative
best docking target/ligand configurations. For the sake of clarity, these
figures show the BSS only for the RBD/ACE2 region highlighted by the yellow
rectangle in Figure 1(b). The following color scheme is adopted for the BSSs:
gray, red, blue, and white for carbon, oxygen, nitrogen, and hydrogen atoms,
respectively. In the ball-stick representation for the flavonoids, the carbon,
oxygen, and hydrogen atoms are shown in the colors cyan, red, and white,
respectively. As a general trend, one can note that the flavonoids fit inside
the core pocket region (cavity) of the RBD/ACE2 complex. This cavity is
displayed as region 2 in Figure 1(a). Acacetin, Luteolin∗, Luteolin∗∗,
Thymusin, and Xanthomicrol were adsorbed on region 1 (see Figure 1(a)) of the
RBD/ACE2 complex. The ligands tend to interact with the oxygen atoms (red
spots in the BSS) in regions 1 and 2. These regions establish pocket-like
media that yield interactions (mostly hydrogen bonds) between flavonoids and
amino acid residues of proteins.

Figures 5 and 6 provide a clear picture of the interaction between the amino
acid residues of the proteins and peppermint flavonoids. The docked poses
(obtained using PLIP salentin2015plip  show the residues names and the bond
types. In the stick representation of flavonoids, the carbon and oxygen atoms
are in the orange and red colors, respectively. The hydrogen, hydrophobic, and
$\pi$-staking bonds are denoted by the blue, dashed gray, and dashed yellow
lines, respectively. The yellow sphere represents the charge center. In Figure
5 one can note that Acacetin, Apigenin, Apigenin∗, Chryseoriol, Hesperidin,
Hesperetin, Ladenein, and Luteolin interact with RBD/ACE2 mainly through 4, 5,
5, 6, 12, 5, 4, and 8 hydrogen bonds with distinct amino acid residues in both
RBD and ACE2 proteins. Similarly, Figure 6 shows the interaction mechanism
between Luteloin∗, Luteloin∗∗, Luteloin∗∗∗, Naringenin, Pebrellin,
Sakuranetin, Thymusin, and Xanthomicrol with RBD/ACE2 is mediated by 7, 5, 9,
8, 5, 5, 4, and 4 hydrogen bonds with distinct amino acid residues in both RBD
and ACE2 proteins, respectively. In total, 12 hydrophobic bonds were found.
The flavonoids and amino acid residues of the proteins involved in this kind
of interaction are highlighted below. Some $\pi$-stacking bonds are also
present in the RBD/ACE2 interactions with flavonoids expecting for the
Hesperidin (Figure 5(e)), Luteolin∗ (Figure 6(a)), and Xanthomicrol (Figure
6(h)) cases.

Generally speaking, we identified 31 distinct amino acid residues of the
RBD/ACE2 interacting with the peppermint flavonoids. The RBD amino acid
residues (and their occurrence) are TYR738 (4), LYS682 (5), GLU761 (6), GLN674
(6), TYR770 (6), ARG688 (8), ASP670 (2), GLY761 (4), GLY741 (2), GLN39 (1),
ALA740 (1), LYS723 (3), ARG673 (1), and SER759 (1). The ACE2 amino acid
residues (and their occurrence) are GLU5 (3), SER1 (5), ASP12 (7), PHE372 (4),
ARG375 (9), ASN15 (8), GLU19 (9), PRO371 (1), ANS15 (1), THR71 (1), ALA369
(4), ARG37 (1), ALA368 (1), LYS335 (2), ASP20 (1), TYR760 (1), and LYS8 (1).
This result suggests that the target RBD/ACE2 amino acid residues for this
class of phytochemicals are ARG375, ASN15, and GLU19 from ACE2, and ARG668
from RBD, based on their higher occurrence. The flavonoids that present
hydrophobic bonds with the RBD/ACE2 amino acids, highlighted in the following
as (flavonoid/residue), are Ladanein/GLU19, Luteolin/LYS682, Hesperetin/ASN15,
Hesperetin/GLU19, Pebrellin/TYR760, Sakuranetin/GLU19, Thymusin/LY58,
Acacetin/GLU5, Apigenin/ASN15, Apigenin/PRO371, Apigenin/TYR770, and
Chryseoriol/LYS682.

##  4 Conclusions

In summary, a set of phytochemicals (peppermint flavonoids) were screened
against the SARS-CoV-2 Spike receptor-binding domain interacting with the
human ACE2 receptor. The approach is based on computationally fitting small
molecules for the target RBD/ACE2 complex proteins using the 3D structure of
the active site with SWISSDOCK grosdidier2011fast ; grosdidier2011swissdock ,
subsequently the ranking of the docked compounds with Quimera
pettersen2004ucsf  and interaction analysis with PLIP salentin2015plip .
Results revealed that Luteolin 7-O-neohesperidoside has a binding affinity of
about -9.18 Kcal/mol, the higher one among the flavonoids studied here. On the
other hand, Sakuranetin was the one with the lowest affinity (about -6.38
Kcal/mol). Binding affinities of the other peppermint flavonoids ranged from
-6.44 Kcal/mol up to -9.05 Kcal/mol. These values outperform the binding
affinities reported by docking studies using other types of compounds in which
the RBD/ACE2 complex was also the target cherrak2020potential ;
bhowmik2021evaluation .

The binding site surface analysis showed pocket-like regions on the RBD/ACE2
complex that yield several interactions (mostly hydrogen bonds) between the
flavonoid and the amino acid residues of the proteins. The interaction
mechanism between the flavonoids and amino acid residues of the proteins is
mediated by hydrogen bonds, essentially. The presence of some hydrophobic and
$\pi-stacking$ bonds was also observed. In total, we identified 31 distinct
amino acid residues of the RBD/ACE2 interacting with the peppermint
flavonoids. The target RBD/ACE2 amino acid residues for this class of
phytochemicals are ARG375, ASN15, and GLU19 from ACE2, and ARG668 from RBD,
based on their higher occurrence.

Some in vitro studies investigated the antiviral activity of flavonoids in
combating SARS-CoV badshah2021antiviral ; RUSSO2020109211  and SARS-CoV2
abian2020structural ; liskova2021flavonoids ; seadawy2021vitro ;
solnier2020flavonoids  infection. Hesperetin, Luteolin, and Apigenin have been
demonstrated as potent inhibitors of SARS-CoV-2 3CLpro in vitro and can be
considered proper candidates for further optimization and development of
therapeutic interventions, particularly those related to inflammation
processes and immunity solnier2020flavonoids . A Luteolin derivative and
Apigenin showed the best docking scores in our study.

###### Acknowledgements.

The authors gratefully acknowledge the financial support from Brazilian
Research Councils CNPq, CAPES, and FAPDF and CENAPAD-SP for providing the
computational facilities. M.L.P.J. gratefully acknowledge the financial
support from CAPES grant 88882.383674/2019-01. L.A.R.J. gratefully
acknowledges respectively, the financial support from FAP-DF grant
00193.0000248/2019-32, CNPq grant 302236/2018-0, and UnB/DPI/DEX Edital
01/2020 grant 23106.057604/2020-05. The molecular graphics and analyses were
performed with UCSF Chimera, developed by the Resource for Biocomputing,
Visualization, and Informatics at the University of California, San Francisco,
with support from NIH P41-GM103311.

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