Content Algebras Over Commutative Rings With Zero-Divisors
Peyman Nasehpour
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15 pages. Note: Since the author has cited to this preprint in some of his papers, it was necessary to update and edit it for the convenience of the readers of his papers
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all $x \in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.
In this article we revisit a problem regarding Bezout domains, namely, whether every Bezout domain is an elementary divisor domain. We prove that a Bezout domain in which every maximal ideal is principal is an elementary divisor ring
A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.
Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different classes of commutative Bezout domains, in particular, $PM^{*}$, local Gelfand domains and so on.
Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acuteΓ_I(R)$ of $R$ and obtain some related results.
Much work has been done on generalized factorization techniques in integral domains, namely $τ$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This paper seeks to synthesize work done in these two areas and extend the notion of $τ$-factorization to commutative rings that need not be domains. In addition, we look into particular types of $τ$ relations, which are interesting when there are zero-divisors present. We then proceed to classify commutative rings that satisfy the finite factorization properties given…
Much work has been done on generalized factorization techniques in integral domains, namely $τ$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. There are many ways authors have decided to study factorization when zero-divisors present. This paper focuses on the method $τ$-complete factorizations developed by D.D. Anderson and A. Frazier. There is a natural way to extend $τ$-complete factorization to commutative rings with zero-divisors. $τ$-complete factorization is a natural way to think of refining factoriza…
In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in the ring, the so called irreducible divisor graph. In this paper, we construct several different associated irreducible divisor graphs of a commutative ring with unity using various choices for the definition of irreducible and atomic in the literature. We continue pursuing the program of exploiting the in…
Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular $τ$-factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing $τ$-factorization to rings with zero-divisors by using the notion of U-factorization.
For a commutative semigroup $S$ with 0, the zero-divisor graph of $S$ denoted by $Γ(S)$ is the graph whose vertices are nonzero zero-divisor of $S$, and two vertices $x$, $y$ are adjacent in case $xy=0$ in $S$. In this paper we study the case where the graph $Γ(S)$ is complete $r$-partite for a positive integer $r$. Also we study the commutative semigroups which are finitely colorable.
Let $R$ be a commutative ring and $Γ(R)$ denote its zero-divisor graph. In this paper, we investigate the genus number of the compact Riemann surface which $Γ(R)$ can be embedded and illustrate all finite commutative rings $R$ (up to isomorphism) such that $Γ(R)$ is either toroidal or planar.
Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely $τ$-factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by D.D. Anderson and S. Valdes-Leon. We investigate how one can extend the notion of $τ$-factorization to commutative rings with zero-divisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct no…
The global defensive $k$-alliance is a very well studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for $k=-1$ known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results.
We investigate eigenvalues of the zero-divisor graph $Γ(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $Γ(R)$. The graph $Γ(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $Γ(R)$, i.e. the multiplicity of the eigenvalue 0 of $Γ(R)$. Moreover, we precisely determine the spectra of $Γ(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $Γ(\mathbb Z_p …
Let $A$ be a commutative augmented ring and $I$ be its augmentation ideal. This paper shows that the sequence $\{I^n/I^{n+1}\}$ becomes stationary up to isomorphism. The result yields stability in the associated graded ring of $A$ along $I$.
Let $R$ be a commutative ring with unity not equal to zero and let $Γ(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the…
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
In this paper we prove that if $R$ is a commutative, reduced, local ring, then $R$ is Hopfian if and only if the ring $R[x]$ is Hopfian. This answers a question of Varadarajan, in the case when $R$ is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.
The compressed zero-divisor graph $Γ_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $Γ_C(R)$ if and only if $xy = 0$ for all $x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Specifically, we prove several theorems which exhibit a method of constructing $Γ(R)$ for when one quotients out by a principal ideal, and prove suf…
In 1988, I.Beck showed that the chromatic number of G(Zn) is equal to its clique number.In 2004, S.Akbari and A.Mohammadian proved that the edge chromatic number of G(Zn) is equal to its maximum degree,in 2008, J.Skowronek-kaziow give formulas calculating the clique number and the maximum degree of G(Zn), but he have a error about clique number of G(Zn), we consider the zero-divisor graph G(Zn) of the ring Zn.we give formulas calculating the clique number of G(Zn).We present a constructed method to calculate the clique number.
We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.
A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian lo…
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+y\in Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
We extend to rings with zero-divisors the concept of perinormal domain introduced by N. Epstein and J. Shapiro. A ring $A$ is called perinormal if every overring of $A$ which satisfies going down over $A$ is $A$-flat. The Prüfer rings and the Marot Krull rings are perinormal.
For an artinian ring $R$, the directed zero-divisor graph $Γ(R)$ is connected if and only if there is no proper one-sided identity element in $R$. Sinks and sources are characterized and clarified for finite ring $R$, especially, it is proved that for {\it any} ring $R$, if there exists a source $b$ in $Γ(R)$ with $b^2=0$, then $|R|=4$ and $R=\{0,a,b,c\}$, where $a$ and $c$ are left identity elements and $ba=0=bc$. Such a ring $R$ is also the only ring such that $Γ(R) $ has exactly one source. This shows that $Γ(R)$ can not be a network for any ring $R$.
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals fo…
Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization domain. If $R$ is not a domain, this is no longer true. In this paper we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.
We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.
In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$ are either bipartite graphs or complete graphs with horns.
Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\…
Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite. We expand upon this result and prove that the reciprocal of the $F$-signature of a local strongly $F$-regular ring $R$ bounds the cardinality of the torsion subgroup of the divisor class group of $R$.
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f(t) in R[t] then there is a factorization of f(t) into irreducibles of length no more than deg(f(t)) + 2.
We shall describe the divisor class group and the graded canonical module of the multi-section ring for a normal projective variety X and Weil divisors D_1,..., D_s on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor due to Hashimoto.
We establish basic results on subrings of finite commutative rings and closely related rings. Among other applications we calculate the number of maximal subrings of a finite commutative local ring.
We study the ring of arithmetical functions with unitary convolution, giving an isomorphism to a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett between the ring of arithmetical functions with Dirichlet convolution and the power series ring on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units.
This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.
In this paper, we introduce and study the $S$-weak global dimension $S$-w.gl.dim$(R)$ of a commutative ring $R$ for some multiplicative subset $S$ of $R$. Moreover, commutative rings with $S$-weak global dimension at most $1$ are studied. Finally, we investigated the $S$-weak global dimension of factor rings and polynomial rings.
Let $R$ be a commutative ring with identity and $T(R)$ its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of commutative rings satisfying this property and to shed light on some open questions raised by Heinzer and Roitman in \cite{HR}.
