# On S-Bézout rings

Mohamed Chhiti  Mohamed Chhiti
Laboratory of Modelling and Mathematical Structures,
Faculty of Economics and Social Sciences of Fez, University S.M. Ben Abdellah
Fez, Morocco.  $E-mail\ address:\<EMAIL_ADDRESS>,  Salah Eddine
Mahdou  Salah Eddine Mahdou
Laboratory of Modelling and Mathematical Structures,
Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben
Abdellah Fez, Morocco.  $E-mail\ address:\<EMAIL_ADDRESS>and
Moutu Abdou Salam Moutui  Moutu Abdou Salam Moutui
Division of Science, Technology, and Mathematics
American University of Afghanistan, Doha, Qatar<EMAIL_ADDRESS>Moutu
Abdou Salam Moutui
University of Haute Alsace, IRIMAS
Département de Mathématiques
18, rue des frères Lumière, F-68093 Mulhouse, France.  abdou-salam-
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce the concept of $S$-Bézout ring, as a
generalization of Bézout ring. We investigate the relationships between
$S$-Bézout and other related classes of rings. We establish some
characterizations of $S$-Bézout rings. We study this property in various
contexts of commutative rings including direct product, localization, trivial
ring extensions and amalgamation rings. Our results allow us to construct new
original classes of $S$-Bézout rings subject to various ring theoretical
properties. Furthermore, we introduce the notion of nonnil $S$-Bézout ring and
establish some characterizations.

###### Key words and phrases:

$S$-Bézout, localization, direct product, $S$-principal ideal, pullback,
trivial ring extension, amalgamation.

###### 2010 Mathematics Subject Classification:

13A15, 13B99, 13E15.

##  1\. Introduction

Throughout this whole paper, all rings are assumed to be commutative with
nonzero identity and all modules are nonzero unital. In the last forty years,
there have been many extensions of classical notions used in multiplicative
ideal theory, with the need of generalizing many properties, including Bézout
domains, to broader contexts. The notion of Bézout ring is central in
commutative algebra and enlarging this class as well as producing new original
examples can be of some interest. In [4], Anderson and Zafrullah extended the
class of Bézout domains in the following way: they called a domain $A$ an
almost Bézout domain if, given any two elements $a,b\in A$, there exists an
integer $n\geq 1$ such that the ideal $(a^{n},b^{n})$ of $A$ is principal.
Later, Mahdou, Mimouni and Moutui enlarged the notion almost Bézout domain to
ring with zero divisors. In recent years, the concept of $S$-property has an
important place in commutative algebra and they draw attention by several
authors. In [5], Anderson and Dumitrescu introduced the concept of $S$-finite
modules, where $S$ is a multiplicatively subset as follows: an $R$-module $M$
is called an $S$-finite module if there exist a finitely generated
$R$-submodule $N$ of $M$ and $s\in S$ such that $sM\subseteq N.$ Also, they
introduced the concept of $S$-Noetherian rings as follows : a ring $R$ is
called $S$-Noetherian if every ideal of $R$ is $S$-finite. Recently, in [14],
Bennis and El Hajoui investigated the $S$-versions of finitely presented
modules and coherent modules which are called, respectively, $S$-finitely
presented modules and S-coherent modules. An $R$-module $M$ is called an
$S$-finitely presented module for some multiplicative closed subset $S$ of $R$
if there exists an exact sequence of $R$-modules $0\to K\to F\to M\to 0$,
where $F$ is a finitely generated free $R$-module and $K$ is an $S$-finite
$R$-module. Moreover, an $R$-module $M$ is said to be $S$-coherent, if it is
finitely generated and every finitely generated submodule of $M$ is
$S$-finitely presented. They showed that the $S$-coherent rings have a similar
characterization to the classical one given by Chase for coherent rings (see
[14, Theorem 3.8]$)$. Any coherent ring is $S$-coherent and any $S$-Noetherian
ring is $S$-coherent. For more detail about $S$-Noetherian and $S$-coherent
rings, we refer the reader to [5, 14].

Let $R$ be a ring and $M$ be an $R$-module. Then $R\propto M$, the trivial
(ring) extension of $R$ by $M$, is the ring whose additive structure is that
of the external direct sum $R\oplus M$ and whose multiplication is defined by
$(r_{1},m_{1})(r_{2},m_{2}):=(r_{1}r_{2},r_{1}m_{2}+r_{2}m_{1})$ for all
$r_{1},r_{2}\in R$ and all $m_{1},m_{2}\in M$. This construction is also known
by other terminology, such as the idealization. See for instance [22, 23].
Trivial ring extensions have an important place in commutative ring theory,
due to their effectiveness of producing new classes of examples and counter
examples of rings subject to various ring theoretic properties (see for
example [1, 6, 13, 19, 22, 23, 24]).

Let $(A,B)$ be a pair of rings, $J$ be an ideal of $B$ and $f:A\rightarrow B$
be a ring homomorphism. The amalgamation of $A$ with $B$ along $J$ with
respect to $f$ is the following subring of $A\times B$:

$A\bowtie^{f}J=\\{(a,f(a)+j)\diagup a\in A,j\in J\\}.$

As a natural generalization of the duplication construction in [18], the
amalgamation ring was initiated by D’Anna, Finocchiaro, and Fontana in [16].
This ring extension covers several ring theoretic constructions, including the
$A+XB[X]$, $A+XB[[X]]$, and the $D+M$ constructions as well as several
classical constructions, such as trivial ring extensions and the CPI
extensions. For more details regarding the amalgamation ring, we refer the
reader to [2, 15, 16, 17, 18, 21].

It is worthwhile recalling that a prime ideal $P$ of $R$ is divided if
$P\subseteq(x)$ for any $x\in R\setminus P$. A ring $R$ is said to be divided
if every prime ideal of $R$ is divided. For more details on divided rings and
related notions, we refer the reader to [12]. Badawi has extensively
investigated in [7, 8, 9, 10, 11], the following class of rings:
$\mathcal{H}=\\{R\mid R$ is a commutative ring and $Nil(R)$ is a divided prime
ideal of $R\\}$. Note that in the case $R\in\mathcal{H}$, $R$ is called a
$\phi$-ring. Obviously, every integral domain is a $\phi$-ring. An ideal $I$
of $R$ is said to be a nonnil ideal if $I\nsubseteq$ $Nil(R)$. If $I$ is a
nonnil ideal of a $\phi$-ring $R$, then $Nil(R)$ $\subseteq I$. Let $R$ be a
ring with total quotient ring $T$ such that $Nil(R)$ is a divided prime ideal
of $R$. As in [8], we define $\phi:T(R)\longrightarrow K:=R_{Nil(R)}$ such
that $\phi(\frac{a}{b})=\frac{a}{b}$ for every $a\in R$ and every $b\in
R\setminus Z(R)$. Then $\phi$ is a ring homomorphism from $T(R)$ into $K$, and
$\phi$ restricted to $R$ is also a ring homomorphism from $R$ into $K$ given
by $\phi(x)=\frac{x}{1}$ for every $x\in R$. Observe that if
$R\in\mathcal{H}$, then $\phi(R)\in\mathcal{H}$, $Ker(\phi)\subseteq Nil(R)$,
$Nil(T(R))=Nil(R)$, $Nil(R_{Nil(R)})=\phi(Nil(R))=Nil(\phi(R))=Z(\phi(R))$,
$T(\phi(R))=R_{Nil(R)}$ is quasilocal with maximal ideal $Nil(\phi(R))$, and
$R_{Nil(R)}/Nil(\phi(R))=T(\phi(R))/Nil(\phi(R))$ is the quotient field of
$\phi(R)/Nil(\phi(R))$.

In this paper, we introduce the concept of $S$-Bézout ring, as a
generalization of Bézout ring. We investigate the relationships between
$S$-Bézout and other related classes of rings. We establish some
characterizations of $S$-Bézout rings. We study this property in various
contexts of commutative rings including direct product, localization, trivial
ring extensions and amalgamation rings. Our results allow us to construct new
original classes of $S$-Bézout rings subject to various ring theoretical
properties. Moreover, we introduce the notion of nonnil $S$-Bézout ring and
establish some characterizations. For a ring $R$, we denote respectively by
$Nil(R)$, $U(R)$, $Z(R)$ the ideal of all nilpotent elements of $R$, the
multiplicative group of units of $R$ and the zero divisor elements of $R$. For
an integral domain $R$, we denote by $qf(R)$, the quotient field of $R$.

##  2\. Main Results

Let $R$ be a ring and let $S$ be a multiplicatively closed subset of $R$.
Recall that a ring $R$ is Bézout if every finitely generated ideal of $R$ is
principal. First, we introduce a weak version of Bézout ring called $S$-Bézout
ring in the following way:

######  Definition 2.1.

Let $R$ be ring and $S$ be a multiplicative set of $R$. Then $R$ is said to be
$S$-Bézout, if every finitely generated ideal of $R$ is $S$-principal.

It is worthwhile noting that any Bézout ring is $S$-Bézout for every
multiplicative set $S$ of $R$. However, the converse is not true in general as
shown in the following examples which illustrate non-Bézout $S$-Bézout rings.

######  Example 2.2.

Let $R:=\mathbb{Z}\propto(\mathbb{Z}/{2\mathbb{Z}})^{\infty}$ be the trivial
ring extension of $\mathbb{Z}$ by the $\mathbb{Z}$-module
$(\mathbb{Z}/{2\mathbb{Z}})^{\infty}$ and $S:=\\{2^{n},0)/n\in\mathbb{N}\\}$
be a multiplicatively set of $R$. Then:

  1. (1)

$R$ is an $S$-Bézout ring.

  2. (2)

$R$ is not a Bézout ring.

Proof.

  1. (1)

Let $J$ be a finitely generated proper ideal of $R$ and set
$I:=\\{a\in\mathbb{Z}/(a,e)\in J$ for some
$e\in(\mathbb{Z}/{2\mathbb{Z}})^{\infty}\\}$. Two cases are then possible:
Case 1. $I\neq 0$.
In this case, say $I=n\mathbb{Z}$, for some $n\in\mathbb{Z}-\\{0\\}$. Let
$e\in(\mathbb{Z}/{2\mathbb{Z}})^{\infty}$ such that $(n,e)\in J$. So, $J$ is
$S$-principal since

$(2,0)J\subseteq R(n,e)\subseteq J$

since $(2,0)J=R(2n,0)=R(n,e)(2,0)\subseteq R(n,e)$, as desired.
Case 2. $I=0$.
Then $J$ is also $S$-principal since

$(2,0)J\subseteq R(0,0)\subseteq J$

since
$(2,0)J\subseteq(2,0)(0\propto(\mathbb{Z}/{2\mathbb{Z}})^{\infty})=R(0,0)$, as
desired.

  2. (2)

Let $(e,f)$ be linearly independant elements of the $\mathbb{Z}$-module
$(\mathbb{Z}/{2\mathbb{Z}})^{\infty}$ and set $J=R(0,e)+R(0,f)$. It is clear
that $J$ is not a principal ideal of $R$, as desired. ∎

######  Example 2.3.

Let $R_{1}$ be a Bézout ring, $R_{2}$ be a non-Bézout ring, $R:=R_{1}\times
R_{2}$ and set $S:=\\{(1,1),(1,0)\\}$ be a multiplicative set of $R$. For
instance take $R_{1}:=K[x]$ the polynomial ring with coefficient in a field
$K$, and $R_{2}:=F[X,Y]$, the polynomial ring in two variables $X,Y$ over a
field $F$. It is well known that $R_{2}$ is a non-Bézout domain. Then:

  1. (1)

$R$ is an $S$-Bézout ring.

  2. (2)

$R$ is a non-Bézout ring.

Proof.

  1. (1)

We claim that $R$ is an $S$-Bézout ring. Indeed, let $I:=I_{1}\times I_{2}$ be
a finitely generated ideal of $R:=R_{1}\times R_{2}$. Hence, $I_{1}$ is a
finitely generated ideal of a Bézout ring $R_{1}$ and so $I_{1}=R_{1}a_{1}$
for some $a_{1}\in R_{1}$. Hence,

$(1,0)(I_{1}\times I_{2})\subseteq I_{1}\times 0(=R(a_{1},0))\subseteq
I_{1}\times I_{2}$

and so $I:=I_{1}\times I_{2}$ is an $S$-principal ideal, as desired.

  2. (2)

$R:=R_{1}\times R_{2}$ is a non-Bézout ring since $R_{2}$ is a non-Bézout
ring.∎

Recall that a ring $R$ is called an $S$-PIR if every ideal is $S$-principal,
that is for every ideal $I$ there exist a principal ideal $J$ and $s\in S$
such that $sI\subseteq J\subseteq I$. Clearly, any $S$-PIR is an $S$-Bézout
ring for any multiplicatively closed subset $S$ of $R$.

######  Example 2.4.

Let $R$ be any non-Bézout domain and set $S:=R-\\{0\\}$ be a multiplicatively
set of $R$. Then:

  1. (1)

$R$ is $S$-PIR. In particular, $R$ is $S$-Bézout ring.

  2. (2)

$R$ is not a Bézout ring.

Proof.

  1. (1)

Let $I$ be a proper ideal of $R$ and let $s\in I-\\{0\\}$. Hence, $sI\subseteq
Rs\subseteq I$ and so $I$ is $S$-principal since $Rs$ is a principal ideal of
$R$, as desired.

  2. (2)

By hypothesis. ∎

Recall that an ideal $I$ is said $n$-generated if it can be generated by $n$
elements. It is well known that a Bézout ring is a ring in which every
$n$-generated (resp., $2$-generated ideal) is principal. Now, we generalize
this result to $S$-Bézout ring.

######  Proposition 2.5.

Let $R$ be ring and $S$ be a multiplicative set of $R$. Then $R$ is an
$S$-Bézout ring if and only if every $2$-generated ideal is $S$-principal.

Proof.  If $R$ is an $S$-Bézout ring, then every $2$-generated ideal is
$S$-principal. Conversely, assume that every $2$-generated ideal is
$S$-principal. It suffices to shows that if every $n$-generated ideal is
$S$-principal, then every $(n+1)$-generated ideal is $S$-principal.
Assume that every $n$-generated ideal is $S$-principal and let
$I=\sum_{i=1}^{n+1}Ra_{i}$ be an $(n+1)$-generated ideal of $R$. Then there
exist $s\in S$ and $a\in R$ such that

$s\sum_{i=1}^{n}Ra_{i}\subseteq Ra\subseteq\sum_{i=1}^{n}Ra_{i}$

by hypothesis since $\sum_{i=1}^{n}Ra_{i}$ is an $n$-generated ideal of $R$.
Using the fact that $sRa_{n+1}\subseteq Ra_{n+1}$, we have

$s\sum_{i=1}^{n+1}Ra_{i}\subseteq s\sum_{i=1}^{n}Ra_{i}+Ra_{n+1}\subseteq
Ra+Ra_{n+1}\subseteq\sum_{i=1}^{n+1}Ra_{i}.$

On the other hand, $Ra+Ra_{n+1}$ is $2$-generated, then by hypothesis, there
exist $s^{\prime}\in S$ and $b\in R$ such that

$s^{\prime}(Ra+Ra_{n+1})\subseteq Rb\subseteq
Ra+Ra_{n+1}(\subseteq\sum_{i=1}^{n+1}Ra_{i}).$

But, we have $ss^{\prime}\sum_{i=1}^{n+1}Ra_{i}\subseteq
s^{\prime}(Ra+Ra_{n+1})$. Therefore,

$ss^{\prime}\sum_{i=1}^{n+1}Ra_{i}\subseteq Rb\subseteq\sum_{i=1}^{n+1}Ra_{i}$

as $ss^{\prime}\in S$, which completes the proof. ∎

The next theorem establishes another characterization of $S$-Bézout ring.

######  Theorem 2.6.

Let $R$ be a ring and $S$ be a multiplicative set of $R$. Then $R$ is an
$S$-Bézout ring if and only if every $S$-finite ideal of $R$ is $S$-principal.

Proof.  If every $S$-finite ideal of $R$ is $S$-principal, then it is clear
that $R$ is $S$-Bézout. Conversely, assume that $R$ is $S$-Bézout and let $I$
be an $S$-finite ideal of $R$. Then there exist a finitely generated ideal $J$
of $R$ and $s\in S$ such that:

$sI\subseteq J\subseteq I.$

Hence, $J$ is $S$-principal and so there exist $s^{\prime}\in S$ and $a\in R$
such that

$s^{\prime}J\subseteq Ra\subseteq J.$

Therefore,

$ss^{\prime}I\subseteq s^{\prime}J\subseteq Ra\subseteq J\subseteq I$

and so $I$ is $S$-principal, as desired. ∎

Next, we investigate the stability of $S$-Bézout ring property under
localization.

######  Proposition 2.7.

Let $R$ be a ring and $S\subseteq R$ be a multiplicative set of $R$. Then:

  1. (1)

If $R$ is an $S$-Bézout ring, then $S^{-1}R$ is a Bézout ring.

  2. (2)

Assume that $R$ is $S$-Noetherian (not necessary Noetherian). Then $R$ is an
$S$-Bézout ring if and only if $S^{-1}R$ is a Bézout ring.

Proof.  $(1)$ Assume that $R$ is $S$-Bézout and let $J$ be a finitely
generated ideal of $S^{-1}R$. There exists a finitely generated ideal $I$ of
$R$ such that $J=S^{-1}I$. Then $I$ is $S$-principal and hence $J:=S^{-1}I$ is
principal, as desired.
$(2)$ If $R$ is an $S$-Bézout ring, then by assertion $(1)$ above, $S^{-1}R$
is a Bézout ring. Conversely, assume that $S^{-1}R$ is a Bézout ring with the
hypothesis that $R$ is $S$-Noetherian (not necessary Noetherian). Then
$S^{-1}R$ is a PIR. Then $R$ is an $S$-PIR by [5, Proposition 2(g)]. Hence,
$R$ is an $S$-Bézout ring, as desired. ∎

Contrary to Proposition 2.7, the following proposition shows that we do not
always need the assumption ”$S$-Noetherian”.

######  Proposition 2.8.

Let $R$ be a ring and $S\subseteq U(R)$ be a multiplicative set of $R$. Then
$R$ is an $S$-Bézout ring if and only if $S^{-1}R$ is a Bézout ring.

If we choose $S\subseteq U(R)$, then we obtain the following known corollary.

######  Corollary 2.9.

Every Noetherian Bézout ring is a PIR.

For a prime ideal $P$ of $R$, we define $R$ to $P$-Bézout if $R$ is
$(R-P)$-Bézout. We establish the following characterization for a semilocal
ring.

######  Theorem 2.10.

For a semilocal ring $R$, the following statements are equivalent:

  1. (1)

$R$ is a Bézout ring.

  2. (2)

$R$ is a $P$-Bézout ring for all prime ideal $P$ of $R$.

  3. (3)

$R$ is a $M$-Bézout ring for all maximal ideal $M$ of $R$.

Proof.  $(1)\Rightarrow(2)$ Straightforward.
$(2)\Rightarrow(3)$ Clear.
$(3)\Rightarrow(4)$ We can use the same proof of [5, Proposition 12]. ∎

The next proposition examines the $S$-Bézout ring property under homomorphic
image.

######  Proposition 2.11.

Let $R$ be a ring, $I$ be a finitely generated ideal of $R$ and $S$ be a
multiplicatively set of $R$. If $R$ is an $S$-Bézout ring, then $R/I$ is an
$(S+I)$-Bézout ring. The converse is true if there is $s_{0}\in S$ such that
$s_{0}I=0$.

Proof.  Assume that $R$ is $S$-Bézout. Let $J/I$ be a finitely generated ideal
of $R/I$, where $J$ is a finitely generated ideal of $R$. Then there exist
$s\in S$ and $a\in R$ such that

$sJ\subseteq Ra\subseteq J$

since $R$ is a Bézout ring. Hence, we have

$(s+I)(J/I)\subseteq(R/I)(a+I)\subseteq J/I$

and so $J/I$ is $(S+I)$-principal. Hence, $R/I$ is $(S+I)$-Bézout. Conversely,
assume that $R/I$ is $(S+I)$-Bézout and there exists $s_{0}\in S$ such that
$s_{0}I=0$. Consider a finitely generated ideal $J$ of $R$. Then $((J+I)/I)$
is a finitely generated ideal of an $(S+I)$-Bézout ring $R/I$. Hence, there
exist $s\in S$ and $a\in R$ such that:

$(s+I)((J+I)/I)\subseteq(R/I)(a+I)\subseteq(J+I)/I.$

By multiplying the above equation by $s_{0}$ and since $s_{0}I=0$, we obtain :

$ss_{0}J\subseteq R(s_{0}a)\subseteq s_{0}J\subseteq J,$

and so $J$ is $S$-principal. Hence, $R$ is an $S$-Bézout ring which completes
the proof. ∎

Now we study the stability of $S$-Bézout property under homomorphism. Let $A$
and $B$ be two rings, $f:A\rightarrow B$ be a ring homomorphism and $I$ be an
ideal of $A$. We denote by $I^{e}$, the extension of $I$ defined by the ideal
of $B$ generated by $f(I)$; and by $J^{c}$ the contraction of $J$ defined by
the set of antecedents of elements of $f(A)\cap J$, that is, $J^{c}=\\{a\in
A|f(a)\in J\\}$.

######  Proposition 2.12.

Let $(A,B)$ be a pair of rings, $f:A\rightarrow B$ be a ring homomorphism and
$S$ be a multiplicative set of $A$ such that $I^{ce}=I$ for each ideal $I$ of
$B$ and $J^{c}$ is a finitely generated ideal of $A$ for each finitely
generated ideal $J$ of $B$. If $A$ is an $S$-Bézout ring, then $B$ is an
$f(S)$-Bézout ring.

Proof.  Assume that $A$ is an $S$-Bézout ring. Let $J$ be a finitely generated
ideal of $B$. From assumption $J^{c}$ is a finitely generated ideal of $A$
which is $S$-Bézout. So, $J^{c}$ is $S$-principal and therefore there exist
$s\in S$ and $a\in A$ such that $sJ^{c}\subseteq aA\subseteq J^{c}$. It
follows that $f(s)J=f(s)J^{ce}\subseteq(aA)^{e}\subseteq J^{ce}=J$. Hence,
$f(s)J\subseteq f(a)B\subseteq J$, making $J$, an $f(S)$-principal ideal of
$B$. Finally, $B$ is an $f(S)$-Bézout ring, as desired.∎

It is clear that if $S_{1},...,S_{n}$ are multiplicative sets of rings
$R_{1},...,R_{n}$ respectively, then $S=\prod_{i=1}^{n}S_{i}$ is a
multiplicative set of $R=\prod_{i=1}^{n}R_{i}$. Next, we study the stability
of the $S$-Bezout property under direct product.

######  Proposition 2.13.

Let $S_{1},...,S_{n}$ be multiplicative sets of rings $R_{1},....,R_{n}$
respectively. Set $R=\prod_{i=1}^{n}R_{i}$ and $S=\prod_{i=1}^{n}S_{i}$ a
multiplicatively closed subset of $R$. The following statements are
equivalent:

  1. (1)

$R$ is an $S$-Bézout ring.

  2. (2)

$R_{i}$ is an $S_{i}$-Bézout ring for all $i=1,...,n$.

Proof.  It suffices to prove the result for $n=2$.
Assume that $R_{1}\times R_{2}$ is an $(S_{1}\times S_{2})$-Bézout ring and
let $I_{i}$ be a finitely generated ideal of $R_{i}$, for $i=1,2$. Then
$I_{1}\times I_{2}$ is a finitely generated ideal of an $(S_{1}\times
S_{2})$-Bézout ring $R_{1}\times R_{2}$. Hence, there exist $s_{i}\in S_{i}$
and $a_{i}\in R_{i}$ such that

$(s_{1},s_{2})(I_{1}\times I_{2})\subseteq(R_{1}\times
R_{2})(a_{1},a_{2})\subseteq I_{1}\times I_{2},$

and so $s_{1}I_{1}\subseteq R_{1}a_{1}\subseteq I_{1}$ and
$s_{2}I_{2}\subseteq R_{2}a_{2}\subseteq I_{2}$. Hence, $R_{i}$ is an
$S_{i}$-Bézout ring, as desired. Conversely, assume that $R_{i}$ is an
$S_{i}$-Bézout ring for each $i=1,2$ and let $I_{1}\times I_{2}$ be a finitely
generated ideal of $R_{1}\times R_{2}$. Hence, there exist $s_{i}\in S_{i}$
and $a_{i}\in R_{i}$ such that $s_{1}I_{1}\subseteq R_{1}a_{1}\subseteq I_{1}$
and $s_{2}I_{2}\subseteq R_{2}a_{2}\subseteq I_{2}$ and so we have

$(s_{1},s_{2})(I_{1}\times I_{2})\subseteq(R_{1}\times
R_{2})(a_{1},a_{2})\subseteq I_{1}\times I_{2}.$

Hence, $R_{1}\times R_{2}$ is an $(S_{1}\times S_{2})$-Bézout ring which
completes the proof. ∎

The next proposition studies the $S$-Bézout property under the ring extension
$A\subseteq B$, where $(A,B)$ is a pair of rings.

######  Proposition 2.14.

Let $A\subseteq B$ be a ring extension such that $IB\cap A=I$ for each ideal
$I$ of $A$ and $S\subseteq A$ a multiplicative set. If $B$ is an $S$-Bézout
ring, then so is $A$.

Proof.  Let $I$ be a finitely generated ideal of $A$. Since $B$ is an
$S$-Bézout ring, then there exist $s\in S$ and $b\in B$ such that
$sIB\subseteq bB\subseteq IB$. Clearly $b\in IB$ and so can be picked from
$I$. Therefore, $sI=sIB\cap A\subseteq bB\cap A\subseteq bA\subseteq I$,
making $I$ an $S$-principal ideal of $A$. Hence, $A$ is an $S$-Bézout ring, as
desired.∎

Let $R:=A\propto E$ be the trivial ring extension of a ring $A$ by an
$A$-module $E$. Note that if $S$ is a multiplicative set of $R$, then
$S_{0}=\\{a\in A/(a,e)\in S$ for some $e\in E\\}$ is a multiplicative set of
$A$. Conversely, if $S_{0}$ is a multiplicative set of $A$, then
$S:=S_{0}\propto N$ is a multiplicative set of $R$ for every submodule $N$ of
$E$ such that $S_{0}N\subseteq N$. In particular, $S_{0}\propto 0$ and
$S_{0}\propto E$ are multiplicative sets of $R$. Our next result examines the
transfer of the $S$-Bézout property in trivial ring extension in the special
setting $E$ is a finitely generated $A$-module.

######  Theorem 2.15.

Let $A$ be a ring, $E$ be a finitely generated $A$-module, $S_{0}$ be a
multiplicatively closed subset of $A$, $R:=A\propto E$ be the trivial ring
extension of $A$ by $E$ and $S:=S_{0}\propto E$ be a multiplicatively closed
subset of $R$. If $R$ is an $S$-Bézout ring, then $A$ is an $S_{0}$-Bézout
ring and every finitely generated submodule $F$ of $E$ is $S$-cyclic (in
particular $E$ is $S$-cyclic). The converse holds if every ideal of $R$ is
homogeneous.

Before proving the previous theorem, we establish the following lemma.

######  Lemma 2.16.

Let $A$ be a ring, $E$ be an $A$-module, $F$ be a submodule of $E$ and $S$ be
a multiplicatively closed subset of $A$. Then $F$ is $S$-cyclic if and only if
$0\propto F$ is an $S\propto E$-principal ideal of $A\propto E$.

Proof.  Assume that $F$ is $S$-cyclic. Then there exist $s\in S$ and $e\in F$
such that

$sF\subseteq Ae\subseteq F$,

and so

$0\propto sF\subseteq 0\propto Ae\subseteq 0\propto F$,

therefore,

$(s,0)0\propto F\subseteq A\propto E(0,e)\subseteq 0\propto F$.

Hence, $0\propto F$ is an $S\propto E$-principal ideal of $A\propto E$.
Conversely, assume that $0\propto F$ is an $S\propto E$-principal ideal of
$A\propto E$. Then there exists $(s,e)\in S\propto E$ such that

$(s,e)0\propto F\subseteq A\propto E(a,e^{\prime})\subseteq 0\propto F$, for
some $(a,e^{\prime})\in A\propto E$,

and so

$0\propto sF\subseteq Aa\propto(Ae^{\prime}+aE)\subseteq 0\propto F$

therefore,

$0\propto sF\subseteq 0\propto Ae^{\prime}\subseteq 0\propto F$.

Hence, $sF\subseteq Ae^{\prime}\subseteq F.$ Finally, $F$ is $S$-cyclic, as
desired. ∎

Proof of Theorem 2.15.  Assume that $R$ is an $S$-Bézout ring. Let $F$ be a
finitely generated submodule of $E$. Then $0\propto F$ is a finitely generated
ideal of $R$ by [1, Theorem 2(1)] and so is $S$-principal. Therefore, $F$ is
$S$-cyclic by Lemma 2.16. On the other hand, consider a finitely generated
ideal $I$ of $A$. From [1, Theorem 7(1)], $I\propto IE$ is a finitely
generated ideal of $R$. Consequently, there exist $(s,e)\in S$ and
$(a,e^{\prime})\in I\propto IE$ such that

$(s,e)I\propto IE\subseteq A\propto E(a,e^{\prime})\subseteq I\propto IE$,

so,

$sI\subseteq Aa\subseteq I$.

Hence, $I$ is $S_{0}$-principal, making $A$, an $S_{0}$-Bézout ring.
Conversely, assume that every ideal of $R$ is homogeneous (recall from [6],
that homogeneous ideals of $R$ have the form $I\propto F$ for some ideal $I$
of $A$ and some submodule $F$ of $E$ with $IE\subseteq F$). Let $I\propto F$
be a finitely generated ideal of $R$. By [1, Theorem 9(1)], $I$ is a finitely
generated ideal of $A$ and $F$ is a finitely generated submodule of $E$. So,
there exist $(s,s^{\prime})\in S_{0}^{2}$, $a\in I$ and $e\in F$ such that

$sI\subseteq Aa\subseteq I$,

and

$s^{\prime}F\subseteq Ae\subseteq F$.

Set $t=ss^{\prime}\in S$, then

$tI\subseteq Aa\subseteq I$

and

$tF\subseteq Ae\subseteq Ae+aE\subseteq F,$

and so

$tI\propto tF\subseteq Aa\propto(Ae+aE)\subseteq I\propto F$,

therefore,

$(t,0)I\propto F\subseteq A\propto E(a,e)\subseteq I\propto F$.

Hence, $I\propto F$ is $S_{0}\propto E$-principal. Finally, $R$ is an
$S$-Bézout ring, as desired. ∎

The following theorem investigates the transfer of $S$-Bézout ring property in
various context of trivial ring extensions.

######  Theorem 2.17.

Let $A$ be a ring, $E$ be an $A-$module, $S_{0}$ be a multiplicatively closed
subset of $A$, $R:=A\propto E$ be the trivial ring extension of $A$ by $E$ and
$S:=S_{0}\propto E$ be a multiplicatively closed subset of $R$. Then the
following statements hold:

  1. (1)

If $R$ is an $S$-Bézout ring, then $A$ is an $S_{0}$-Bézout ring.

  2. (2)

Assume that $A$ is an integral domain which is not a field, $K=qf(A)$, and
$R:=A\propto K$ be the trivial ring extension of $A$ by $K$. Then $R$ is an
$S$-Bézout ring if and only if $A$ is an $S_{0}$-Bézout domain.

  3. (3)

Assume that $(A,M)$ is a local ring and $E$ is an $A-$module such that $ME=0$
and $S_{0}\nsubseteq U(R)(=R-(M\propto E)$. Then $R$ is an $S$-Bézout ring if
and only if $A$ is an $S_{0}$-Bézout ring.

Proof.  $(1)$ Assume that $R$ is an $S$-Bézout ring and let $I$ be a finitely
generated proper ideal of $A$ generated by $(a_{i})_{i=1,\ldots,n}$. Then the
finitely generated ideal $J$ of $R$ generated by $(a_{i},0)_{i=1,\ldots,n}$ is
$S$-principal, that is there exist $(s_{0},f)\in S$ and $(a,e)\in R$ such that

$(s_{0},f)J\subseteq R(a,e)\subseteq J$

and so

$s_{0}I\subseteq Aa\subseteq I.$

Therefore, $I$ is an $S_{0}$-principal and hence $A$ is an $S_{0}$-Bézout
ring.

$(2)$ If $R$ is an $S$-Bézout ring, then $A$ is an $S_{0}$-Bézout domain by
assertion $(1)$ above. Conversely, let $J$ be a finitely generated proper
ideal of $R$. Set $I:=\\{a\in A/(a,e)\in J$ for some $e\in K\\}$. Two cases
are then possible:
Case 1. $I=0$.
Necessarily, $J=0\propto(1/b)L$ for some $b\not=0\in A$ and some finitely
generated proper ideal $L$ of $A$. Therefore, $L$ is an $S_{0}$-principal
since $A$ is an $S_{0}$-Bézout ring and so there exist $s_{0}\in S_{0}$ and
$a\in A$ such that

$s_{0}L\subseteq Aa\subseteq L$

and we have

$(s_{0},0)J\subseteq R(0,a/b)\subseteq J.$

Hence, $J$ is an $S$-principal ideal of $R$, as desired.
Case 2. $I\not=0$.
Let $(a,e)\in J$ such that $a\not=0$. Then $(a,e)(0\propto K)=0\propto
K\subseteq J$; equivalently, $J=I\propto IK=I\propto K$, where $I$ is a
finitely generated ideal of an $S_{0}$-Bézout ring $A$. Hence, there exist
$s_{0}\in S_{0}$ and $a\in A$ such that

$s_{0}I\subseteq Aa\subseteq I,$

and so we have

$(s_{0},0)J\subseteq R(a,0)\subseteq J.$

Hence, $J$ is an $S$-principal ideal, as desired.
$(3)$ If $R$ is an $S$-Bézout ring, then A is an $S_{0}$-Bézout ring by
assertion $(1)$ above. Conversely, assume that $A$ is an $S_{0}$-Bézout ring
and $S_{0}\nsubseteq U(R)(=R-(M\propto E)$, and let $J$ be a finitely
generated ideal of $R$. Set $I:=\\{a\in A$ such that $(a,e)\in J$ for some
$e\in E\\}$ which is a finitely generated ideal of $A$. Since $A$ is an
$S_{0}$-Bézout ring, then there exist $s_{0}\in S_{0}$ and $a\in A$ such that

$s_{0}I\subseteq Aa\subseteq I.$

By multiplying the above equation by an element $s\in S_{0}\cap M$, we may
assume that $s_{0}\in M$ (since we have $ss_{0}I\subseteq Asa\subseteq
sI\subseteq I$). On the other hand, there exists $e\in E$ such that $(a,e)\in
J$ since $a\in I$. Hence, $(as_{0},0)=(a,e)(s_{0},0)\in J$ and so we have

$(s_{0}s_{0},0)J\subseteq R(s_{0},0)(a,e)=R(s_{0}a,0)\subseteq J.$

Therefore, $J$ is an $S$-principal ideal of $R$, which completes the proof.
$\Box$

Next, we use the transfer of the $S$-Bézout property in the trivial ring
extension to provide some new original examples of $S$-Bézout rings that are
not Bézout.

######  Example 2.18.

Let $(A,P)$ be a local Bézout domain, $E=A/P$ be an $A$-module such that
$Dim_{\frac{A}{P}}(E)\neq 1$ and let $S_{0}$ be a multiplicatively closed
subset of $A$ such that $S_{0}\cap P\neq\emptyset$. Consider the trivial ring
extension of $A$ by $E$ $R:=A\propto E$ and $S:=S_{0}\propto E$ a
multiplicatively closed subset of $R$. Then:

  1. (1)

$R$ is an $S$-Bézout ring.

  2. (2)

$R$ is not a Bézout ring.

Proof.  $(1)$ Observe that $PE=0$ and $S_{0}\not\subseteq U(R)$. By assertion
$(3)$ of Theorem 2.17, $R$ is an $S$-Bézout ring (as $A$ is an $S_{0}$-Bézout
ring).
$(2)$ Assume by the way of contradiction that $R$ is a Bézout ring. Since
$Dim_{\frac{A}{P}}(E)\neq 1$, then $Dim_{\frac{A}{P}}(E)\geq 2$. Pick two
elements $h,k\in E$ such that {h,k} is $A/P$-linearly independant set and
consider the finitely generated ideal $J:=R(0,h)+R(0,k)$ of $R$. One can
easily check $J$ is not principal, which is a contradiction. Therefore, $R$ is
not a Bézout ring.∎
We denote by $\overline{H}$, the closure integral of integral domain $H$ in
its quotient field. Recall from [4] that an integral domain $H$ is almost
Bézout if and only if the integral closure $\overline{H}$ of $H$ is a Prüfer
domain with torsion class group and $H\subseteq\overline{H}$ is a root
extension. Now, we show how one can build an example of an $S$-Bézout almost
Bézout ring which is not Bézout via trivial ring extension of an integral
domain by a vector space over its quotient field, as shown below.

######  Example 2.19.

Let $F$ be a field of characteristic $p>0$ and let $F\subseteq L$ be a purely
inseparable field extension. Consider the integral domain $H=F+XL[X],$
$K:=qf(A)$, $R:=H\propto K$ the trivial ring extension of $H$ by $K$ and
$S:=S_{0}\propto E$ be a multiplicative closed subset of $R$, where
$S_{0}:=H-\\{0\\}$ is a multiplicative closed subset of $H$. Then:

  1. (1)

$R$ is an $S$-Bézout ring.

  2. (2)

$R$ is an almost Bézout ring.

  3. (3)

$R$ is not a Bézout ring.

Proof.

$(1)$ By assertion $(2)$ of Theorem 2.17, $R$ is an $S$-Bézout ring, as $H$ is
an $S_{0}$-Bézout ring.
$(2)$ Note that $\overline{H}=L[X]$ is a Prüfer domain with torsion class
group and for each $Q$ in $L[X]$ there exists $n\geq 0$ such that
$Q^{p^{n}}\in H.$ Therefore, $H\subset\overline{H}$ is a root extension. From
[4, Corollary 4.8(1)], $H$ is an almost Bézout domain and so is $R$ [25,
Theorem 3.1(2)].
$(3)$ $R$ is not a Bézout ring since $H$ is a homomorphic image of $R$ which
is not Bézout (as $H$ is not integrally closed).∎

Theorem 2.17 provides a new example of $S$-Bézout ring which is not Bézout, as
shown below.

######  Example 2.20.

Let $A$ be a non-Bézout integral domain which is not a field, $K=qf(A)$, and
$R:=A\propto K$ be the trivial ring extension of $A$ by $K$. Set
$S_{0}=A-\\{0\\}$ and $S:=S_{0}\propto E$. Then $R$ is an non-Bézout
$S$-Bézout ring by the assertion $(2)$ Theorem 2.17 (as $A$ is a non-Bézout
$S_{0}$-Bézout ring).

Now, we turn our attention to the transfer of the $S$-Bézout ring property to
amalgamation of rings $R:=A\bowtie^{f}J$. It is worthwhile observing that if
$S^{\prime}$ is a multiplicative closed subset of $R$, then $S_{0}=\\{a\in
A/(a,f(a)+j)\in S^{\prime}$ for some $j\in J\\}$ is multiplicative closed
subset of $A$. Conversely, if $S_{0}$ is a multiplicative set of $A$, then
$S^{\prime}:=S_{0}\bowtie^{f}0$ and $S_{0}\bowtie^{f}J$ are multiplicative
sets of $R$.

######  Theorem 2.21.

Let $A$ and $B$ be two rings, $J$ an ideal of $B$ and let $f:A\longrightarrow
B$ be a ring homomorphism. Then:

  1. (1)

If $A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring, then $A$ is an
$S_{0}$-Bézout ring.

  2. (2)

Assume that $f(S_{0})\cap J\neq\emptyset$. Then $A\bowtie^{f}J$ is an
$S^{\prime}$-Bézout ring if and only if $A$ is an $S_{0}$-Bézout ring.

  3. (3)

Assume that $f(S_{0})\cap J=\emptyset$, $J:=ann(f(S_{0}))$ and every proper
ideal of $A\bowtie^{f}J$ is homogenous. Then $A\bowtie^{f}J$ is an
$S^{\prime}$-Bézout ring if and only if $A$ is an $S_{0}$-Bézout ring.

Proof.  $(1)$ Assume $R:=A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring and let
$I$ be a finitely generated proper ideal of $A$ generated by
$(a_{i})_{i=1,\ldots,n}$. Then the finitely generated ideal $J$ of $R$
generated by $(a_{i},f(a_{i}))_{i=1,\ldots,n}$ is $S$-principal, that is there
exist $(s_{0},f(s_{0})+j)\in S$ and $(a,f(a)+j^{\prime})\in R$ such that

$(s_{0},f(s_{0})+j)J\subseteq R(a,f(a)+j^{\prime})\subseteq J$

and so

$s_{0}I\subseteq Aa\subseteq I.$

Therefore, $I$ is $S_{0}$-principal and hence $A$ is an $S_{0}$-Bézout ring.
$(2)$ Assume that $f(S_{0})\cap J\neq\emptyset$. If $A\bowtie^{f}J$ is an
$S^{\prime}$-Bézout ring, then by assertion $(1)$ above, $A$ is an
$S_{0}$-Bézout ring. Conversely, assume that $A$ is $S_{0}$-Bézout. Let $L$ be
a finitely generated ideal of $A\bowtie^{f}J$. Consider the ideal $I:=\\{a\in
A/(a,f(a)+j)\in L$ for some $j\in J\\}$ of $A$. Since $L$ is a finitely
generated proper ideal of $A\bowtie^{f}J$, then it is easy to see that $I$ is
a finitely generated proper ideal of $A$. Using the fact that $A$ is an
$S_{0}$-Bézout ring, then $I$ is an $S_{0}$-principal ideal of $A$. So, there
exist $s_{0}\in S_{0}$ and $a\in A$ such that $s_{0}I\subseteq aA\subseteq I$.
From assumption, there is $t\in S_{0}$ such that $f(t)\in J$. Observe that
$ts_{0}I\subseteq atA\subseteq tI\subseteq I$. Therefore,
$(ts_{0},0)L\subseteq ts_{0}I\times 0\subseteq atA\times
0\subseteq(a,f(a))tA\times 0\subseteq(a,f(a))A\bowtie^{f}J\subseteq L$, since
$f(t)\in J$ and $a\in I$. Hence, $L$ is an $S^{\prime}$-principal ideal of
$A\bowtie^{f}J$, making $A\bowtie^{f}J$, an $S^{\prime}$-Bézout ring.
$(3)$ Assume that $f(S_{0})\cap J=\emptyset$, $J:=ann(f(S_{0}))$ and every
proper ideal of $A\bowtie^{f}J$ is homogenous. If $A\bowtie^{f}J$ is an
$S$-Bézout ring, then by assertion $(1)$ above, $A$ is an $S_{0}$-Bézout ring.
Conversely, assume that $A$ is an $S_{0}$-Bézout ring. Let $L$ be a finitely
generated proper ideal of $A\bowtie^{f}J$. From assumption, $L:=I\bowtie^{f}J$
for some finitely generated proper ideal $I$ of $A$. Since $A$ is an
$S_{0}$-Bézout ring, then there exist $s_{0}\in S_{0}$ and $a\in A$ such that
$s_{0}I\subseteq aA\subseteq I$. We claim that
$(s_{0},f(s_{0}))L\subseteq(a,f(a))A\bowtie^{f}J\subseteq L$. Indeed, let
$(i,f(i)+k)\in L$. Then $s_{0}i=ar$ for some $r\in A$. So,
$(s_{0},f(s_{0}))(i,f(i)+k)=(s_{0}i,f(s_{0}i)+kf(s_{0}))=(s_{0}i,f(s_{0}i))=(ar,f(ar))=(a,f(a))(r,f(r))\in(a,f(a))A\bowtie^{f}J\subseteq
L$. Hence, it follows that $I\bowtie^{f}J$ is an $S^{\prime}$-principal ideal
of $A\bowtie^{f}J$. Finally, $A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring,
as desired.∎

Next, let $I$ be a _proper_ ideal of $A$. The (amalgamated) duplication of $A$
along $I$ is a special amalgamation given by

$A\bowtie I:=A\bowtie^{id_{A}}I=\big{\\{}(a,a+i)/a\in A,i\in I\big{\\}}.$

The following corollary is an immediate consequence of Theorem 2.21 on the
transfer of the $S$-Bézout property into duplications.

######  Corollary 2.22.

Let $A$ be a ring and $I$ be an ideal of $A$. Then:

  1. (1)

If $A\bowtie I$ is an $S^{\prime}$-Bézout ring, then $A$ is an $S_{0}$-Bézout
ring.

  2. (2)

Assume that $S_{0}\cap I\neq\emptyset$. Then $A\bowtie I$ is an
$S^{\prime}$-Bézout ring if and only if so is $A$ is an $S_{0}$-Bézout ring.

  3. (3)

Assume that $S_{0}\cap I=\emptyset$, $I:=Ann(f(S_{0}))$ and every proper ideal
of $A\bowtie I$ is homogenous. Then $A\bowtie I$ is $S^{\prime}$-Bézout if and
only if $A$ is an $S_{0}$-Bézout ring.

Theorem 2.21 enriches the current literature with a new original class of
$S$-Bézout rings which are not Bézout. The following examples show how to
construct such rings.

######  Example 2.23.

Let $A$ be any $S$-Bézout ring which is not Bézout (for instance take
$A:=B\propto E$ be the trivial ring extension of $B$ by $E$, $B:=\mathbb{Z}$
be the ring of integers, $E:=(\mathbb{Z}/2\mathbb{Z})^{\infty}$ be a
$\mathbb{Z}/2\mathbb{Z}$-vector space, see example 2.2). Consider the
surjective ring homomorphism $f:A\rightarrow B$ and
$S_{0}:=\\{(2^{n},0)/n\in\mathbb{N}\\}$ be a multiplicative set of $A$ and
$J:=2\mathbb{Z}$ be an ideal of $B$. Then:

  1. (1)

$A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.

  2. (2)

$A\bowtie^{f}J$ is not a Bézout ring.

Proof.  $(1)$ First, observe that $f(S_{0})\cap J\neq\emptyset$. Since $A$ is
an $S_{0}$-Bézout ring (by Example 2.2), then by assertion $(2)$ of Theorem
2.21, it follows that $A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.
$(2)$ We claim that $A\bowtie^{f}J$ is not a Bézout ring. Indeed, $A$ which is
a homomorphic image of $A\bowtie^{f}J$ is not Bézout and the fact that the
Bézout property is stable under factor ring, it follows that $A\bowtie^{f}J$
is not a Bézout ring.∎

######  Example 2.24.

Let $A:=A_{1}\times A_{2}$ be a ring (for instance $A:=\mathbb{Z}\times
K[X,Y]$ where $\mathbb{Z}$ is the ring of integers and $K[X,Y]$ the polynomial
ring in two variables $X,Y$ over a field $K$). Consider the multiplicative
sets $S_{1}:=\\{3^{n}/n\in\mathbb{N}\\}$ and $S_{2}:=K[X,Y]-\\{0\\}$ of
$\mathbb{Z}$ and $K[X,Y]$, respectively, the surjective ring homomorphism
$f:\mathbb{Z}\times K[X,Y]\rightarrow\mathbb{Z}$ defined by $f((a,P))=a$ and
the ideal $J:=3\mathbb{Z}$ of $\mathbb{Z}$. Let $S=S_{1}\times S_{2}$ be a
multiplicative set of $A$ and $S^{\prime}:=S\bowtie^{f}0$ be a multiplicative
set of $A\bowtie^{f}J$. Then:

  1. (1)

$A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.

  2. (2)

$A\bowtie^{f}J$ is not a Bézout ring.

Proof.  $(1)$ First note that $A:=\mathbb{Z}\times K[X,Y]$ is an $S_{1}\times
S_{2}$-Bézout ring (by Proposition 2.13, since $\mathbb{Z}$ is an
$S_{1}$-Bézout ring and $K[X,Y]$ is an $S_{2}$-Bézout ring). On the other
hand, $f(S)\cap J=S_{1}\cap J\neq\emptyset$. Hence, by assertion $(2)$ of
Theorem 2.21, $A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.
$(2)$ $A\bowtie^{f}J$ is not a Bézout ring, since $A$ is not a Bézout ring (as
$K[X,Y]$ is not a Bézout domain).∎

The next example shows that the class of $S$-Bézout rings and the class of
almost Bézout rings are distinct in general and we use Theorem 2.21 to
construct a new original class of $S$-Bézout rings which are not almost
Bézout.

######  Example 2.25.

Let $A$ be any integral domain which is not a field, $f=id_{A}$ be the
identity ring homomorphism and let $J:=m$ be a maximal ideal of $A$. Consider
the multiplicative closed subset $S_{0}:=A-\\{0\\}$ of $A$. Then:

  1. (1)

$A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.

  2. (2)

$A\bowtie^{f}J$ is not an almost Bézout ring.

Proof.  $(1)$ Using similar argument as in Example 2.4, it follows that $A$ is
an $S^{\prime}$-Bézout ring. On the other hand $f(S_{0})=f(A-\\{0\\})\cap
J=A-\\{0\\}\cap m\neq\emptyset$. Hence, by assertion $(2)$ of Theorem 2.21, it
follows that $A\bowtie^{f}J$ is an $S^{\prime}$-Bézout ring.
$(2)$ From [26, Corollary 2.5], $A\bowtie^{f}J$ is not an almost Bézout ring,
as $J\neq 0$.∎

Now, we introduce the concept of nonnil $S$-Bézout ring.

######  Definition 2.26.

Let $A\in\mathcal{H}$ be a ring and $S\subseteq A$ a multiplicative set. $A$
is called a nonnil $S$-Bézout ring if for any pair of finitely generated
nonnil ideals $I$ and $P$ of $A$ such that $I\subseteq P$ and $P$ is an
$S$-principal ideal, we have $I$ is $S$-principal ideal.

Next, we establish a characterization of nonnil $S$-Bézout property. For every
multiplicative set $S$ of ring $A$, set
$S^{\prime}=\frac{S}{Nil(A)}=\\{s+Nil(A)\mid s\in S\\}$. It is easy to see
that $S^{\prime}$ is a multiplicative set of $\frac{A}{Nil(A)}$.

######  Theorem 2.27.

Let $A\in\mathcal{H}$ be a ring and $S\subseteq A$ a multiplicative set. Then
$A$ is a nonnil $S$-Bézout ring if and only if $\frac{A}{Nil(A)}$ is an
$S^{\prime}$-Bézout domain.

Proof. Assume that $A$ is a nonnil $S$-Bézout ring and let
$\frac{I}{Nil(A)}\subseteq\frac{P}{Nil(A)}$ be a pair of finitely generated
ideals of $\frac{A}{Nil(A)}$ such that $\frac{P}{Nil(A)}$ is an
$S^{\prime}$-principal ideal of $\frac{A}{Nil(A)}$. We may assume that
$\frac{I}{Nil(A)}\neq 0$. Hence, $I\subseteq P$ is a pair of finitely
generated nonnil ideals of $A$. Therefore, there exist $s^{\prime}\in
S^{\prime}$ and $\bar{a}\in\frac{A}{Nil(A)}$ such that
$s^{\prime}\frac{P}{Nil(A)}\subseteq\bar{a}\frac{A}{Nil(A)}\subseteq\frac{P}{Nil(A)}$
with $s^{\prime}=s+Nil(A)$ for some $s\in S$. So,
$\bar{a}\frac{A}{Nil(A)}=(ab+Nil(A))$ for some $b\in A$. We have
$\bar{a}\frac{A}{Nil(A)}\cong\frac{J}{Nil(A)}$. Our aim is to show that $J$ is
a principal ideal of $A$ generated by $a$. Consider a nonnilpotent element $x$
of $J$. Then $x+Nil(A)=ab+Nil(A)$ in $\frac{A}{Nil(A)}$ for some $b$ in $A$.
Therefore, there exists $w\in Nil(A)$ such that $x+w=ab$ in $A$. Since $x$ is
nonnilpotent, $w=xk$ for some $k\in Nil(A)$. So, $x+w=x+xk=x(1+k)=ab$. Using
the fact that $k\in Nil(A)$ and so $1+k\in U(A)$, it follows that $x\in aA$
and so $J$ is a principal ideal of $A$ generated by $a$. One can easily check
that $sP\subseteq aA=J\subseteq P$ and so $P$ is an $S$-principal ideal of
$A$. Consequently, $I$ is $S$-principal ideal of $A$ as $A$ is a nonnil
$S$-Bézout ring. It follows that $\frac{I}{Nil(A)}$ is an
$S^{\prime}$-principal ideal of $\frac{A}{Nil(A)}$. Therefore,
$\frac{A}{Nil(A)}$ is an $S^{\prime}$-Bézout domain. Conversely, let
$I\subseteq P$ be a pair of finitely generated nonnil ideals of $A$ such that
$P$ is an $S$-principal ideal of $A$. Then
$\frac{I}{Nil(A)}\subseteq\frac{P}{Nil(A)}$ is a pair of nonzero finitely
generated ideals of $\frac{A}{Nil(A)}$ and it is easy to see that
$\frac{P}{Nil(A)}$ is an $S^{\prime}$-principal ideal of $\frac{A}{Nil(A)}$.
Therefore, $\frac{I}{Nil(A)}$ is $S^{\prime}$-principal since
$\frac{A}{Nil(A)}$ is an $S^{\prime}$-Bézout domain. Finally, $I$ is
$S$-principal, making $A$, a nonnil $S$-Bézout ring. ∎

As a consequence of Theorem 2.27, we establish the following characterization
of nonnil $S$-Bézout rings.

######  Corollary 2.28.

Let $A\in\mathcal{H}$ be a ring and $S\subseteq A$ be a multiplicative set.
Then $A$ is a nonnil $S$-Bézout ring if and only if $\phi(A)$ is a nonnil
$\phi(S)$-Bézout ring.

Proof. Assume that $A$ is a nonnil $S$-Bézout ring. By Theorem 2.27,
$\frac{A}{Nil(A)}$ is an $S^{\prime}$-Bézout domain with
$S^{\prime}=\frac{S}{Nil(A)}$. From [11, Lemma 2.1],
$\frac{A}{Nil(A)}\cong\frac{\phi(A)}{Nil(\phi(A))}$ and therefore
$\frac{\phi(A)}{Nil(\phi(A))}$ is a $\frac{\phi(S)}{Nil(\phi(S))}$-Bézout
domain. Hence, $\phi(A)$ is a nonnil $\phi(S)$-Bézout ring by Theorem 2.27.
The converse holds using similar argument as previously.∎

The following example illustrates Theorem 2.27 by generating a new original
class of nonnil $S$-Bézout ring which is not Bézout.

######  Example 2.29.

Let $A$ be any non-Bézout $S$-Bézout domain which is not a field with quotient
field $K$ and let $S\subseteq A$ be a multiplicative set. Let $K$ be the
quotient field of $A$. Then $A\propto K$ is a nonnil $S$-Bézout ring which is
not Bézout.

Proof. Set $R=A\propto K$. First, $Nil(R)=0\propto K$ is a divided prime ideal
of $R$. Indeed, let $(0,e)\in Nil(R)$ and $(a,f)\in R\setminus Nil(R)$. Then
$(0,e)=(a,f)(0,\frac{e}{a})$ and thus $R\in\mathcal{H}$. Since
$\frac{R}{Nil(R)}$ is ring-isomorphic to $A$, $R$ is a nonnil $S$-Bézout ring
by Theorem 2.27. Furthermore, $R$ is not a Bézout ring.∎

The following theorem establishes a characterization of nonnil $S$-Bézout
rings in a special setting of pullback.

######  Theorem 2.30.

Let $A\in\mathcal{H}$ and $S\subseteq A$ a multiplicative set. Then $A$ is a
nonnil $S$-Bézout ring if and only if $\phi(A)\cong R$, where $R$ is obtained
from the following pullback diagram:

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T/M}$

where $T$ is a zero-dimensional quasilocal ring with maximal ideal $M$,
$B:=R/M$ is a $S_{1}$-Bézout subring of $T/M$ with $S_{1}=\alpha(\phi(S))/M$
such that $\alpha$ is the ring isomorphism from $\phi(A)$ to $R$, the vertical
arrows are the usual inclusion maps, and the horizontal arrows are the usual
surjective maps.

Proof. Assume that $\phi(A)\cong R$ obtained from the given diagram. Then
$R\in\mathcal{H}$ and $Nil(R)=Z(R)=M$. Since $R/M$ is an $S_{1}$-Bézout
domain, $R$ is a nonnil $S_{2}$-Bézout ring by Theorem 2.27, where
$S_{2}=\alpha(\phi(S))$, and so $\phi(A)$ is a nonnil $\phi(S)$-Bézout ring.
Hence, $A$ is a nonnil $S$-Bézout ring. Conversely, assume that $A$ is a
nonnil $S$-Bézout ring. Set $T=A_{Nil(A)}$, $M=Nil(A_{Nil(A)})$, and
$R=\phi(A)$ yields the desired pullback diagram. ∎

The next theorem establishes a result showing that the class of nonnil Prüfer
rings and the class of nonnil $P$-Bézout rings coincide. Recall that an
integral domain $R$ is local Bézout if and only if it is a valuation domain.

######  Theorem 2.31.

Let $R\in\mathcal{H}$. Then the following assertions are equivalent:

  1. (1)

$R$ is a nonnil $P$-Bézout ring for each prime ideal $P$ of $R$.

  2. (2)

$R_{P}$ is a nonnil chained ring for each prime ideal $P$ of $R$.

  3. (3)

$R_{M}$ is a nonnil chained ring for each maximal ideal $M$ of $R$.

  4. (4)

$R$ is a nonnil Prüfer ring.

Proof.  $(1)\Rightarrow(2)$ Assume that $R$ is a nonnil $P$-Bézout ring for
each prime ideal $P$ of $R$. Let $P$ be a prime ideal of $R$. Then $R$ is a
nonnil $R-P$-Bézout ring. By Theorem 2.27, $\frac{R}{Nil(R)}$ is an
$\frac{R-P}{Nil(R)}$-Bézout domain and so by assertion $(1)$ of Proposition
2.7, $\left(\frac{R}{Nil(R)}\right)_{\frac{P}{Nil(R)}}$ is a local Bézout
domain and so is a valuation domain. On the other hand,
$\left(\frac{R}{Nil(R)}\right)_{\frac{P}{Nil(R)}}\simeq\frac{R}{Nil(R)R_{P}}\simeq\frac{R_{P}}{Nil(R_{P})}$
and $R_{P}\in\mathcal{H}$. Therefore, by [3, Theorem 2.7], it follows that
$R_{P}$ is a nonnil chained ring.
$(2)\Rightarrow(1)$ Assume that $R_{P}$ is a nonnil chained ring for each
prime ideal $P$ of $R$. Observe that $R_{P}\in\mathcal{H}$ and by [3, Theorem
2.7],
$\left(\frac{R}{Nil(R)}\right)_{\frac{P}{Nil(R)}}\simeq\frac{R_{P}}{Nil(R_{P})}$
is a valuation domain and so is a local Bézout domain. So, by Theorem 2.10,
$\frac{R_{P}}{Nil(R_{P})}$ is
$(\frac{R_{P}}{Nil(R_{P})}-\frac{PR_{P}}{Nil(R_{P})})$-Bézout. Using the fact
that $(\frac{R_{P}}{Nil(R_{P})}-\frac{PR_{P}}{Nil(R_{P})})\subseteq
U(\frac{R_{P}}{Nil(R)})$, then by Proposition 2.8, it follows that
$\frac{R}{Nil(R)}$ is a $\frac{P}{Nil(R)}$-Bézout domain and so by Theorem
2.27, $R$ is a nonnil $P$-Bézout ring, as desired.
$(2)\Leftrightarrow(3)\Leftrightarrow(4)$ This follows from [3, Theorem 2.9].∎

## References

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# Robustness of Solar-Cycle Empirical Rules Across Different Series Including
an Updated ADF Sunspot Group Series

Ilya Usoskin  Gennady Kovaltsov  Wilma Kiviaho  University of Oulu, Finland
St.Petersburg State University, Russia  A.F. Ioffe Physical-Technical
Institute, St. Petersburg, Russia  University of Turku, Finland

###### keywords:

Solar Cycle, Observations; Sunspots, Statistics

##  1 Introduction

S:Intro

The series of sunspot number is the longest and probably the most analyzed
astrophysical dataset based on consistent direct telescopic observations since
1610 (e.g. Clette et al., 2014; Usoskin, 2017). It provides a quantitative
index of solar magnetic variability, which is dominated by the
$\approx$11-year Schwabe cycle. However, this cyclic variability also exhibits
variations of individual cycles both in strength/magnitude and the length (see
a detailed review by Hathaway, 2015). This includes also the Maunder minimum
(1645 – 1715) when sunspots activity was dramatically suppressed (Eddy, 1976;
Usoskin et al., 2015) and the Modern grand maximum in the second half of the
20th century (Solanki et al., 2004; Lockwood et al., 2018).

Because of the length of the sunspot series, solar cyclic variability has been
extensively studied including several empirical rules relating the temporal
evolution of the solar cycles with their strengths. Here we consider the
following empirical rules. One is the Waldmeier rule (Waldmeier, 1935, 1939)
which relates the duration of the ascending phase of a solar cycle to its
amplitude. The other is the Gnevyshev–Ohl rule (Gnevyshev and Ohl, 1948)
(sometimes called the Even–Odd effect), which implies a strong connection
between the integral intensities of an even-numbered and the subsequent odd-
numbered cycles. These rules are empirical, they are based on an analysis of
the data and they serve as important observational constraints for solar
dynamo models (e.g. Charbonneau, 2020).

For more than a century, the sunspot series, initially compiled by Rudolf Wolf
and continued by his successors in Zürich, called the Wolf sunspot series
(WSN), remained the “golden truth” forming a basis for numerous studies. The
WSN series is composed via a linearly scaled combination of the counts of
sunspot groups and individual spots reported by different observers with
different skills and instrumentation (see details in Hathaway, 2015). This WSN
series was continued as the International sunspot series (ISN) by the Royal
Observatory of Belgium since 1981 (Clette et al., 2007). However, the approach
of R. Wolf in compiling the WSN was questioned by Hoyt and Schatten (1998),
who proposed that the use of only the number of sunspot groups is more robust
and revised the entire set of the individual observers’ data, proposing an
alternative Group sunspot number (GSN). It was shown by Hathaway, Wilson, and
Reichmann (2002) that the use of the GSN series slightly alters the empirical
relations obtained for the WSN/ISN series.

As demonstrated several years ago, the “classical” WSN and GSN series contain
a number of inconsistencies or errors that compromise the reliability of the
data before the onset of the 20th century (e.g. Clette et al., 2014; Usoskin,
2017; Muñoz-Jaramillo and Vaquero, 2019). Moreover, new data of historical
observations have been found and analyzed (e.g. Lockwood, Owens, and Barnard,
2014; Vaquero, 2007; Arlt et al., 2013; Arlt and Vaquero, 2020; Hayakawa et
al., 2020; Carrasco et al., 2020). As a result, several updates and revisions
of the sunspot (Clette and Lefèvre, 2016; Chatzistergos et al., 2017) and
group (Cliver and Ling, 2016; Svalgaard and Schatten, 2016; Usoskin et al.,
2016; Willamo, Usoskin, and Kovaltsov, 2017) number series have been recently
provided.

The empirical rules were studied by different authors (e.g. Usoskin, Mursula,
and Kovaltsov, 2001, 2002; Hathaway, Wilson, and Reichmann, 2002; Aparicio,
Vaquero, and Cruz Gallego, 2012; Tlatov, 2013; Hathaway, 2015; Carrasco et
al., 2016; Takalo and Mursula, 2018) using different sunspot series, WSN and
GSN, and they were not tested systematically.

Thus, there is a need to revisit the empirical rules based on the sunspot
series and check their robustness against the exact dataset used. This forms
the main focus of this work.

##  2 Sunspot Series

S:data For the analysis we selected sunspot series that fulfil the following
criteria:

  * •

The series should extend back until 1749, viz. being of the same length as the
original WSN. Although some series extend back to 1610, we do not consider the
period before 1749 because of the very uncertain data between 1720 – 1740 and
the grand-minimum state of solar activity during the Maunder minimum (1645 –
1715).

  * •

The series should have at least monthly resolution. Annual resolution is
insufficient to analyze the Waldmeier rule.

Four series satisfy these criteria. These series are shown in Figure Fig:data.

Figure 1.: Time series of the sunspot (group) numbers used as described in
Section S:data.Fig:data

  1. 1.

The “classical” WSN, which was the basis for most of the previous analyzes of
the empirical rules, was taken as a monthly series for January 1749 through
May 2015 according to the ISN version 1.0 dataset. It was taken from SILSO
(www.sidc.be/silso/datafiles) and is referred further as ISN-v1.

  2. 2.

The revised ISN includes recent updates and corrections and is referred to as
ISN version 2.0 (Clette and Lefèvre, 2016). Its was taken, as a monthly series
for January 1749 through August 2020, from SILSO and is referred further as
ISN-v2.

  3. 3.

The ‘classical’ GSN series by Hoyt and Schatten (1998) is further referred as
GSN-HS. We used the monthly series covering the time from January 1749 through
December 1995, as downloaded from SILSO.

  4. 4.

A new method of composing the GSN series, called the Active Day Fraction (ADF)
method, was proposed recently (Usoskin et al., 2016; Willamo, Usoskin, and
Kovaltsov, 2017). Here we have updated it by exploiting the renewed database
of individual observer’s data (haso.unex.es/haso/index.php/on-line-
archive/data/) (Vaquero et al., 2016). The details of the update are presented
in Appendix App:A. It is hereafter referenced as the GSN-ADF series. This new
series supersedes the previous ADF series (Usoskin et al., 2016; Willamo,
Usoskin, and Kovaltsov, 2017).

All series were reduced to the overlapping period including Cycles 1 – 22
(1755–1996) for the sake of consistency. Some series contain data gaps in the
earlier part. The gaps shorter than seven months were interpolated by 13-month
running mean values, while two longer gaps in the GSN-ADF series (19 months in
June 1810 – December 1811, and 17 months in November 1814 – March 1816) were
interpolated using a 41-month parabolic fit (similar to Mursula, Usoskin, and
Kovaltsov, 2001).

Another type of GSN reconstructions, based on the original (Svalgaard and
Schatten, 2016) or improved (Chatzistergos et al., 2017) “backbone” method,
provide only annual resolution and are not analyzed here.

##  3 Cycle Parameters

S:cycle Here we are interested in such parameters of the solar cycle as its
phasing (ascending, descending phase and total length), magnitude, and
intensity (cumulative sum of sunspot number over the cycle).

While some community-accepted dates of solar-cycle minima and maxima are
listed for WSN (ftp.ngdc.noaa.gov/space-weather/solar-data/solar-
indices/sunspot-numbers/cycle-data/table_cycle-dates_maximum-minimum.txt) and
ISN (www.sidc.be/silso/cyclesminmax) series, definitions of the cycle maxima
and minima are not unambiguous and may involve information going beyond the
sunspot counts (see discussion in Hathaway, 2015). The often-used method of
the 13-month running-mean smoothing sometimes leads to an unstable definition
as it is affected by short bursts of activity (e.g., Carrasco et al., 2016).

Figure 2.: Time series of the ISN-v2 for Solar Cycle 14. The original monthly
sunspot numbers are shown with the thin black line, while 13-month unweighted
running mean and the low-pass (0.25 yr-1) filtered series are shown with the
thick red and blue lines, respectively.Fig:min

An example is shown in Figure Fig:min for Cycle 14 as presented in the ISN-v2
series. One can see that the raw monthly data exhibit a strong variability up
to 70 %, particularly around solar maximum. A traditional way to define the
dates of cycle maxima and minima refers to the extrema of the 13-month running
mean, viz. the 13-month boxcar in the time domain with (the Gleissberg filter)
or without halving the weight for the edge point, as depicted by the red curve
in Figure Fig:min. However, this curve also depicts strong variability leading
to partly ambiguous definitions, since the moving average is a very poor low-
pass filter, due to its slow roll-off and poor stopband attenuation. It has a
broad frequency response with many sidelobes so that high-frequency
variability leaks into the low-frequency range. For example, the peak of the
13-month smoothed series in early 1906 corresponds to a deep drop in sunspot
number. The definition of the minima is not unambiguous either: the minimum of
Cycle 14 can be either in mid-1901 or in early 1902 as set by random
fluctuations of sunspot occurrence. The minimum of Cycle 15 is even less clear
as it can be early 1912 or late 1914.

Table 1.: Times (as fractional years) of the solar cycle minima and maxima for Cycles 1 – 22, defined as described in Section S:cycle for the four sunspot (group) series analyzed. Cycle numbering is the standard one.Tab:min # | ISN-v1 | ISN-v2 | GSN-HS | GSN-ADF
---|---|---|---|---
| min | max | min | max | min | max | min | max
1 | 1755.46 | 1761.38 | 1755.54 | 1761.38 | 1755.63 | 1760.46 | 1756.22 | 1760.63
2 | 1766.38 | 1770.05 | 1766.29 | 1770.04 | 1766.13 | 1770.05 | 1766.47 | 1769.88
3 | 1775.54 | 1778.88 | 1775.54 | 1778.88 | 1775.63 | 1779.38 | 1775.47 | 1778.97
4 | 1784.63 | 1788.13 | 1784.71 | 1788.13 | 1784.55 | 1788.13 | 1784.55 | 1788.05
5 | 1798.21 | 1803.71 | 1798.13 | 1803.71 | 1798.54 | 1802.05 | 1798.13 | 1802.38
6 | 1809.63 | 1816.46 | 1809.63 | 1816.46 | 1809.21 | 1817.29 | 1809.97 | 1815.88
7 | 1822.54 | 1830.29 | 1822.38 | 1830.38 | 1822.05 | 1830.29 | 1824.88 | 1830.22
8 | 1833.79 | 1837.29 | 1833.79 | 1837.29 | 1833.79 | 1837.21 | 1833.80 | 1837.22
9 | 1843.88 | 1848.46 | 1843.79 | 1848.63 | 1843.54 | 1848.88 | 1843.47 | 1848.97
10 | 1856.04 | 1860.38 | 1856.04 | 1860.29 | 1856.13 | 1860.46 | 1856.13 | 1860.47
11 | 1867.21 | 1871.04 | 1867.21 | 1871.04 | 1867.13 | 1871.13 | 1867.05 | 1871.05
12 | 1878.29 | 1883.54 | 1878.29 | 1883.54 | 1878.29 | 1883.96 | 1878.30 | 1883.97
13 | 1889.21 | 1893.79 | 1889.29 | 1893.71 | 1888.96 | 1893.88 | 1889.13 | 1893.88
14 | 1901.71 | 1906.12 | 1901.71 | 1906.13 | 1901.62 | 1906.20 | 1901.72 | 1906.22
15 | 1912.54 | 1917.87 | 1912.63 | 1917.88 | 1912.71 | 1917.87 | 1912.72 | 1917.80
16 | 1923.45 | 1927.62 | 1923.46 | 1927.71 | 1923.45 | 1927.54 | 1923.38 | 1927.63
17 | 1933.54 | 1938.04 | 1933.63 | 1938.04 | 1933.46 | 1938.04 | 1933.55 | 1937.97
18 | 1944.29 | 1948.12 | 1944.21 | 1948.04 | 1944.21 | 1948.04 | 1944.13 | 1948.13
19 | 1954.12 | 1958.12 | 1954.13 | 1958.13 | 1954.04 | 1958.20 | 1954.05 | 1958.22
20 | 1964.71 | 1969.04 | 1964.71 | 1969.04 | 1964.62 | 1969.12 | 1964.55 | 1969.05
21 | 1976.29 | 1980.37 | 1976.29 | 1980.46 | 1976.12 | 1980.29 | 1976.05 | 1980.22
22 | 1986.20 | 1990.29 | 1986.29 | 1990.38 | 1986.20 | 1990.62 | 1986.38 | 1990.47

Accordingly, we used a low-pass filter (rectangular boxcar in the frequency
domain) with the cutoff frequency 0.25 yr-1, corresponding to a four-year
period (similar to Mursula, Usoskin, and Kovaltsov, 2001). This low-pass curve
(blue in Figure Fig:min) is smooth, with cycle extrema defined unambiguously
irrespectively to the high-frequency noise. Henceforth, we use the low-pass
filtered curves to define the cycle maxima and minima as listed in Table
Tab:min for all the analyzed series. The dates of the cycle minima are well
constrained and agree within 4 – 5 months between series, with the sole
exception of Cycle 7, where the difference between GSN-HS (1822.05) and GSN-
ADF (1824.88) dates is 34 months. The dates for the cycle maxima typically
agree within six months between different series, except for Cycles 5 (20
months between ISN and GSN-HS series) and 6 (17 months between GSN-ADF and
GSN-HS series).

We accordingly re-defined the quantitative parameters of the solar cycle
strength for each cycle and each analyzed series. First, we considered the
magnitude of the cycle [$S_{\rm max}$] as the maximum of the low-pass filtered
curve, which gives a more robust value defined by the whole cycle evolution
rather than by the very “local” activity bursts (see Figure Fig:min). The
values are collected in Table Tab:max.

Table 2.: Maximum smoothed sunspot (group) numbers $S_{\rm max}$ and the integral sum $I$ (see Section S:cycle for the definition) for Cycles 1 – 22 for the four analyzed sunspot (group) series. Cycle numbering is the standard one.Tab:max | ISN-v1 | ISN-v2 | GSN-HS | GSN-ADF
---|---|---|---|---
# | $S_{\rm max}$ | $I$ | $S_{\rm max}$ | $I$ | $S_{\rm max}$ | $I$ | $S_{\rm max}$ | $I$
1 | 73.1 | 5623 | 119.6 | 9361 | 4.6 | 359 | 6.6 | 533
2 | 108.0 | 6451 | 179.1 | 10762 | 8.3 | 509 | 9.4 | 638
3 | 149.6 | 7385 | 250.0 | 12325 | 6.2 | 383 | 11.6 | 706
4 | 142.6 | 10098 | 238.3 | 16807 | 7.8 | 596 | 10.7 | 846
5 | 50.2 | 3425 | 84.5 | 5714 | 3.2 | 192 | 3.5 | 282
6 | 44.4 | 2810 | 74.4 | 4672 | 2.2 | 154 | 5.7 | 446
7 | 66.4 | 4761 | 110.1 | 7936 | 4.9 | 342 | 5.0 | 358
8 | 135.6 | 7852 | 225.7 | 13077 | 8.9 | 504 | 8.0 | 479
9 | 118.3 | 8283 | 204.4 | 14786 | 7.0 | 534 | 7.1 | 569
10 | 98.9 | 6546 | 188.1 | 12437 | 7.0 | 456 | 7.8 | 523
11 | 127.9 | 7483 | 213.4 | 12504 | 7.9 | 478 | 9.6 | 621
12 | 67.7 | 4551 | 113.3 | 7597 | 4.9 | 328 | 5.7 | 409
13 | 88.5 | 5598 | 146.9 | 9322 | 7.8 | 483 | 8.2 | 554
14 | 62.0 | 4436 | 102.6 | 7400 | 5.2 | 381 | 5.3 | 409
15 | 93.5 | 5342 | 155.2 | 8900 | 8.4 | 491 | 7.8 | 474
16 | 77.1 | 4963 | 128.8 | 8277 | 6.9 | 464 | 6.9 | 465
17 | 117.6 | 7220 | 195.1 | 12011 | 10.2 | 641 | 10.3 | 653
18 | 152.4 | 9065 | 219.0 | 13268 | 12.0 | 717 | 11.7 | 720
19 | 206.4 | 11476 | 291.3 | 16274 | 15.6 | 869 | 14.3 | 826
20 | 111.9 | 8412 | 159.1 | 11940 | 8.7 | 690 | 8.5 | 679
21 | 163.0 | 9949 | 232.5 | 13932 | 13.0 | 829 | 12.0 | 784
22 | 163.9 | 9420 | 219.8 | 12673 | 13.3 | 769 | 12.2 | 730

We have also calculated the integral intensity (cumulative sum of monthly
sunspot/group numbers from one cycle minimum to the next one) of solar cycles,
using the different series, as summarized in Table Tab:max.

##  4 Analysis of Empirical Relations

S:analysis Here we overview the empirical relations for the for analyzed
series.

###  4.1 Waldmeier Rule

The classical Waldmeier rule refers to a highly significant correlation
between the length of the ascending phase (from minimum to maximum) [$T_{\rm
A}$] and the magnitude of the cycle [$S_{\rm max}$]. Sometimes, a simplified
Waldmeier rule is used relating the magnitude of the cycle to its full length,
but it is much weaker (Usoskin and Mursula, 2003; Hathaway, 2015). Also, a
relation between the length of a cycle and the magnitude of the subsequent
cycle (called the $n+1$ rule) was found to be strong and statistically
significant (Solanki et al., 2002) as expected from the dynamo theory
(Charbonneau and Dikpati, 2000; Charbonneau, 2020).

Using the dates of the cycle minima and maxima (Table Tab:min) and magnitude
[$S_{\rm max}$] (Table Tab:max) we have estimated the Waldmeier rule for the
four analysed series for Solar Cycles 1 – 22, using the three formulations
described above. The relations for the classical Waldmeier rule are depicted
in Figure Fig:Wald and summarized in Table Tab:rule.

Figure 3.: Visualization of the classical Waldmeier rule with the scatter
plots of the magnitude (Table Tab:max) vs. length of the ascending phase
$T_{\rm A}$ (minimum to maximum) of solar cycles (individual dots) for the
four analyzed sunspot series. The dotted lines depict the best-fit linear
regression lines. Correlation coefficients and the significance are listed in
Table Tab:rule Fig:Wald

The length of the ascending phase varies substantially, ranging between 40 and
100 months. It is interesting that weak Cycles 6 and 7 are characterised by
exceptionally long ascending phases (80 – 100 months) in three of the analyzed
series (Figure Fig:Walda – c) probably affecting the overall Waldmeier rule.
On the other hand, the GSN-ADF exhibits a more compact spread of the $T_{\rm
A}$ values (42 – 72 months), making Cycles 6 and 7 non-outliers.

Table 3.: The Pearson’s linear correlation coefficients $r$ and their significance ($p-$values) corresponding to the Waldmeier rule in different formulations (see text) for the four analyzed series. Tab:rule Waldmeier rule | ISN-v1 | ISN-v2 | GSN-HS | GSN-ADF
---|---|---|---|---
Classical | $r$ | $-0.68_{-0.11}^{+0.14}$ | $-0.73_{-0.09}^{+0.13}$ | $-0.44_{-0.16}^{+0.20}$ | $-0.54_{-0.14}^{+0.18}$
| $p$ | $<0.001$ | $<0.001$ | 0.015 | 0.009
Simplified | $r$ | $-0.35_{-0.18}^{+0.22}$ | $-0.29_{-0.20}^{+0.24}$ | $-0.32_{-0.18}^{+0.22}$ | $-0.27_{-0.20}^{+0.25}$
| $p$ | 0.07 | 0.12 | 0.10 | 0.14
(n+1) | $r$ | $-0.66_{-0.12}^{+0.16}$ | $-0.71_{-0.09}^{+0.13}$ | $-0.42_{-0.17}^{+0.21}$ | $-0.59_{-0.12}^{+0.18}$
| $p$ | $<0.001$ | $<0.001$ | 0.015 | $<0.001$

It can be concluded that the Waldmeier rule appears stable against the exact
data series. The classical rule is found highly significant ($p<0.01$) for
both versions of the ISN and GSN-ADF series, and significant ($p=0.015$) for
the GSN-HS series. The simplified rule is insignificant for all of the
analyzed series ($p>0.05$). The $(n+1)$ rule appears significant ($p=0.015$)
for the GSN-HS series and highly significant ($p<0.01$) for other series.

###  4.2 Gnevyshev–Ohl Rule

The Gnevyshev-Ohl (GO) rule says that the solar cycles tend to be grouped in
pairs of even- and the following odd-numbered cycles so that the odd cycle is
stronger (in the sense of the integral cycle-accumulated intensity [$I$]) than
the even one. The numbering of the cycles follows the standard one introduced
by R. Wolf (see Table Tab:min). The GO rule for the four analyzed series is
shown in Figure Fig:GO.

Figure 4.: Illustration of the GO rule for the four series analyzed. The
$y$-axes values correspond to the cycle-integrated intensities (TableTab:max).
The open and filled circles denote the even and odd-numbered (in the standard
numbering) cycles, respectively, connected in even-odd pairs. The valid GO
rule corresponds to the positively sloped line segments, while negatively
sloped segments indicate breaks of the rule.Fig:GO

For the “classical” ISN-v1 series (Figure Fig:GOa), the GO rule is valid since
the Dalton minimum (Cycles 6 – 21), but it breaks before that. The revised
ISN-v2 series (panel b) displays a similar pattern, viz. the GO rule breaks in
the entrance to the Dalton minimum. The GO rule also fails for the cycle pair
22 – 23. Additionally, the GO rule is nearly broken for the cycle pair 10 –
11, when the odd cycle is stronger than the even cycle by only 0.5 %. The GO
rule of the GSN-HS series (panel c) is similar to that in the ISN-v1 series
since the Dalton minimum but inverts the sign for the cycle pair 2 – 3. The
GSN-ADF series (panel d) is similar to other series, in the sense of the GO
rule, but it exhibits an inverted sign for the cycle pair 6 – 7, during the
Dalton minimum. This may be related to the fact that the ADF method tends to
overestimate solar activity during/around grand minima (Willamo, Usoskin, and
Kovaltsov, 2018).

In summary, the GO rule appears robust against the sunspot series used for the
period after the Dalton minimum (after Cycle 8), while all series imply the
breaking of the rule in the beginning of the Dalton minimum. The ISN-v2 series
leads to almost a break of the GO for Cycles 10 – 11.

The physical cause of the GO rule is not understood (Hathaway, 2015) and this
empirical rule provides an observational constraint to the dynamo theory. In
particular, its interpretation in terms of the relic solar magnetic field
(Mursula, Usoskin, and Kovaltsov, 2001) was disproved. The break of the GO
rule across the Dalton minimum can originate from three possible sources: i)
poor quality of data across the Dalton minimum with large uncertainties of the
reconstructed sunspot (group) numbers, leading to the ‘lost’ solar cycle in
the very end of the 18th century (Usoskin, Mursula, and Kovaltsov, 2001;
Usoskin et al., 2009); ii) phase catastrophe in solar cyclic evolution (e.g.,
Vitinsky, Kopecky, and Kuklin, 1986; Kremliovsky, 1994); iii) random
coincidence, since the GO is not a strict law but a statistical tendency.
However, the fact that the GO rule fails also for the cycle pair 22 – 23
(Komitov and Bonev, 2001) just prior to the modern low Cycle 24, favours the
second option of the phase catastrophe accompanying a quick change of solar
activity from a high to a low level.

##  5 Conclusions

We have tested the robustness of empirical rules related to the evolution of
the 11-year Schwabe solar cycle against recent updates and modifications of
the sunspot (group) series. Four series with monthly temporal resolution have
been analyzed for the period between 1749 and 1996 (Solar Cycles 1 – 22): the
classical Wolf sunspot number series (ISN-v1); the revised Intentional sunspot
number series (ISN-v2); the original group sunspot number series (GSN-HS); and
an updated sunspot group number based on the ADF method (GSN-ADF). A more
robust method (low-pass filter with the cutoff frequency of 0.25 year-1) was
used for definition of the cyclic evolution with respect to the traditional
13-month moving average method.

The Waldmeier rule have been analyzed in three formulations (Table Tab:rule):

  * •

Classical formulation (the magnitude of a cycle is inversely related to the
length of its ascending phase) appears significant for the GSN-HS series and
highly significant for the other three series.

  * •

Simplified formulation (the magnitude of a cycle is inversely related to its
entire length) is insignificant for all the series.

  * •

(n+1) formulation (the magnitude of a cycle is inversely related to the total
length of the preceding cycle) was found significant for the GSN-HS series and
highly significant for the other three series.

Thus, the Waldmeier rule in its classical and (n+1) formulations is robust and
independent of the exact sunspot (group) series (see also Aparicio, Vaquero,
and Cruz Gallego, 2012).

The Gnevushev–Ohl rule (solar cycles are paired in even–odd pairs so that the
odd cycle is more intense than the preceding even cycle) was found robust for
all series for Cycles 8 – 21 (although it nearly fails for Cycles 10 – 11 for
the ISN-v2 series), but unstable across the Dalton minimum and before it. The
GO rule also breaks for the recent Cycles 22 – 23.

These empirical rules form important observational constraints for the solar
dynamo theory (e.g., Charbonneau, 2020), but we leave aside theoretical
interpretations focusing merely on the statistical results and their
(in)dependence on the exact series used. The result implies that the Waldmeier
rules can be considered as very robust, while the Gnevyshev–Ohl rule is robust
only after Solar Cycle 8 and becomes unstable before that.

#### Acknowledgments

This work was partially supported by the Academy of Finland (project No.
321882 ESPERA) and by the Russian Science Foundation (RSF project No.
20-67-46016). The authors benefited from discussions within the ISSI
International Team work #417 (Recalibration of the Sunspot Number Series) and
ISWAT-COSPAR S1-01 team.

##  Appendix A Update of the Sunspot Group Numbers Using the ADF method

App:A The active-day fraction (ADF) method of the sunspot (group) number
reconstruction was invented by Usoskin et al. (2016) and greatly upgraded by
Willamo, Usoskin, and Kovaltsov (2017). It is based on the assumption that the
‘quality’ of a solar observer is entirely defined by his/her observational
threshold, rather than a scaling factor used in other methods. An important
advantage of the ADF method is that it calibrates the observers directly to
the reference series avoiding the daisy-chain calibration intrinsic to other
methods. As a drawback, the ADF method tends to strongly overestimate the low-
activity and moderately underestimate the high-activity cycles (Willamo,
Usoskin, and Kovaltsov, 2018).

After the publications of the ADF-based reconstruction by Willamo, Usoskin,
and Kovaltsov (2017), based on the sunspot group observation database of Hoyt
and Schatten (1998), an essential update was made with the upgrade and
systematisation of all known sunspot records, as collected in a database of
Vaquero et al. (2016). Accordingly, we have revisited the ADF-based sunspot
group number reconstructed here, following exactly the algorithm described by
Willamo, Usoskin, and Kovaltsov (2017), but using the new sunspot number
database (Vaquero et al., 2016). As the reference observer, we used the Royal
Greenwich Observatory (solarscience.msfc.nasa.gov/greenwch.shtml), similarly
to Willamo, Usoskin, and Kovaltsov (2017).

Figure 5.: a) Monthly sunspot group number reconstructed here using the ADF
method and the updated solar observation database (Vaquero et al., 2016).
Uncertainties ($\pm 1\sigma$) are shown in grey. Tabulated values are provided
in the Electronic Supplementary Materials. b) Ratio of the annual values of
the SGN obtained here to those from Willamo, Usoskin, and Kovaltsov (2017).
The unity value is shown by the dashed line. The ratio values are not shown
for the periods around solar cycle minima (SNG$<$1). Fig:ADF

The resultant data-series for the period of 1749 – 1996 is shown in Figure
Fig:ADFa and tabulated in the Electronic Supplementary Materials as monthly
and annual sunspot group series. The ratio of the (annual) series obtained
here to that of Willamo, Usoskin, and Kovaltsov (2017) is shown in Figure
Fig:ADFb. It is kept close (within 10 %) to unity after 1880, with small
trembling being related to the Monte-Carlo procedure of the observers’
calibration. The ratio can, however, vary by up to 40 % before the Dalton
minimum. On the other hand, the ratio is consistent with the unity within the
error bars.

This series supersedes the previous ADS-based SGN (Usoskin et al., 2016;
Willamo, Usoskin, and Kovaltsov, 2017) and is used here for further analysis.

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