Nonzero Ideals Research Articles

Zariski‐Like Topology on S‐Quasi‐Primary Ideals of a Commutative Ring

Let R be a commutative ring with nonzero identity and, S⊆R be a multiplicatively closed subset. An ideal P of R is called an S‐quasi‐primary ideal if P∩S = ∅ and there exists an (fixed) s ∈ S and whenever ab ∈ P for a, b ∈ R then either or In this paper, we construct a topology on the set QPrimS(R) of all S‐quasi‐primary ideals of R which is a generalization of the S‐prime spectrum of R. Also, we investigate the relations between algebraic properties of R and topological properties of QPrimS(R) like compactness, connectedness and irreducibility.

S-Zariski topology on S-spectrum of modules

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, first we give some relations between S-prime and S-maximal submodules that are generalizations of prime and maximal submodules, respectively. Then we construct a topology on the set of all S-prime submodules of M , which is generalization of prime spectrum of M. We investigate when SpecS(M) is T0 and T1-space. We also study on some continuous maps and irreducibility on SpecS(M). Moreover, we introduce the notion of S-radical of a submodule N of M and use it to show the irreducibility of S-variety VS(N).

Weakly J-ideals of commutative rings

Let R be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly J-ideals as a new generalization of J-ideals. We call a proper ideal I of a ring R a weakly J-ideal if whenever a,b ? R with 0 ? ab ? I and a ? J(R), then b ? I. Many of the basic properties and characterizations of this concept are studied. We investigate weakly J-ideals under various contexts of constructions such as direct products, localizations, homomorphic images. Moreover, a number of examples and results on weakly J-ideals are discussed. Finally, the third section is devoted to the characterizations of these constructions in an amalgamated ring along an ideal.

On $ mj $-clean ring and strongly $ mj $-clean ring

In this paper, we introduce the concepts of $ mj $-clean and strongly $ mj $-clean rings which are generalizations of $ j $-clean ring and strongly $ j $-clean ring, respectively. Let $ R $ be a ring with a nonzero identity and $ m\geq 2 $ a positive integer. We call the ring $ R $ as $ mj $-clean if each element of $ R$ can be written as a sum of an $ m $-potent and an element of $J(R)$ and also if these elements are commute, then we call $R$ as strongly $ mj $-clean ring. We examine the algebraic properties of these new concepts and show the effects of these structures on matrix rings, polynomial rings, power series and the transitions between them.

On the arithmetic of monoids of ideals

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero ideals of polynomial rings with at least two indeterminates over noetherian domains. Among others, we show that they are not transfer Krull but they share several arithmetical phenomena with Krull monoids having infinite class group and prime divisors in all classes.

A graph associated to a commutative semiring

Let R R be a commutative finite semiring with nonzero identity and H H be an arbitrary multiplicatively closed subset R R . The generalized identity-summand graph of R R is the (simple) graph G H ( R ) GH(R) with all elements of R R as the vertices, and two distinct vertices x x and y y are adjacent if and only if x + y ∈ H x+y∈H . In this paper, we study some basic properties of G H ( R ) GH(R) . Moreover, we characterize the planarity, chromatic number, clique number and independence number of G H ( R ) GH(R) .

The triple zero graph of a commutative ring

Let $R$ be a commutative ring with non-zero identity. We define the set of triple zero elements of $R$ by $TZ(R)=\{a\in Z(R)^{\ast}:$ there exists $b,c\in R\backslash\{0\}$ such that $abc=0$, $ab\neq0$, $ac\neq0$, $bc\neq0\}.$ In this paper, we introduce and study some properties of the triple zero graph of $R$ which is an undirected graph $TZ\Gamma(R)$ with vertices $TZ(R),$ and two vertices $a$ and $b$ are adjacent if and only if $ab\neq0$ and there exists a non-zero element $c$ of $R$ such that $ac\neq0$, $bc\neq0$, and $abc=0$. We investigate some properties of the triple zero graph of a general ZPI-ring $R,$ we prove that $diam(TZ\Gamma(R))\in\{0,1,2\}$ and $gr(G)\in\{3,\infty\}$.

Functional equations related to higher derivations in semiprime rings

Abstract We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.

Revisiting G-Dedekind domains

AbstractLet R be an integral domain with $qf(R)=K$ , and let $F(R)$ be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if, for each $I\in F(R)$ , the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori, and Krull domains. In addition, we show that a Schreier DCD is a greatest common divisor (GCD) (Greatest Common Divisor) domain with the property that, for each $A\in F(R)$ , the ideal $A_{v}$ is principal. We show that a domain R is G-Dedekind (i.e., has the property that $A_{v}$ is invertible for each $A\in F(R)$ ) if and only if R is a DCD satisfying the property $\ast :$ For all pairs of subsets $\{a_{1},\ldots ,a_{m}\},\{b_{1},\ldots ,b_{n}\}\subseteq K\backslash \{0\}, (\cap _{i=1}^{m}(a_{i})(\cap _{j=1}^{n}(b_{j}))=\cap _{i,j=1}^{m,n}(a_{i}b_{j})$ . We discuss what the appropriate names for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

Injective envelopes and the intersection property

We consider the ideal structure of a reduced crossed product of a unital C∗-algebra equipped with an action of a discrete group. More specifically we find sufficient and necessary conditions for the group action to have the intersection property, meaning that non-zero ideals in the reduced crossed product restrict to non-zero ideals in the underlying C∗-algebra. We show that the intersection property of a group action on a C∗-algebra is equivalent to the intersection property of the action on the equivariant injective envelope. We also show that the centre of the equivariant injective envelope always contains a C∗-algebraic copy of the equivariant injective envelope of the centre of the injective envelope. Finally, we give applications of these results in the case when the group is C∗-simple.

Domains whose ideals meet a universal restriction

Let [Formula: see text] represent a set of proper nonzero ideals [Formula: see text] (respectively, t-ideals [Formula: see text]) of an integral domain [Formula: see text] and let P be a valid property of ideals of D. We say S(D) meets P (denoted [Formula: see text] if each [Formula: see text] is contained in an ideal satisfying P. If S(D) [Formula: see text] [Formula: see text] cannot be controlled. When [Formula: see text] [Formula: see text] [Formula: see text] does not imply I(R) [Formula: see text] while [Formula: see text] [Formula: see text] implies [Formula: see text] [Formula: see text] usually. We say S(D) meets P with a twist [Formula: see text]written [Formula: see text] if each [Formula: see text] is such that, for some [Formula: see text] [Formula: see text] is contained in an ideal satisfying P and study [Formula: see text] as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

Reprint of: Endomorphism rings of reductions of Drinfeld modules

Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let πp∈Ep be the Frobenius endomorphism of ϕ⊗Fp. Then we have the inclusion of orders A[πp]⊂Ep⊂Op in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(Ep/A[πp]) and m divides the index χ(Op/Ep). We show that the index χ(Ep/A[πp]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r=2 case we describe an algorithm for computing the orders A[πp]⊂Ep⊂Op, and give some computational data.

On 1-absorbing δ-primary ideals

Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

1-absorbing primary submodules

Abstract Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N : RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

K-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form {a1,a2,a3,…,ak}, where a1,a2,a3,…,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of {a1,a2,a3,…,ak} are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.

Structure of rings with commutative factor rings for some ideals contained in their centers

This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ is commutative for some proper ideal $I$ of $R$ with $I\subseteq Z(R)$, where $Z(R)$ is the center of $R$. We prove that (i) for a CIFC ring $R$, $W(R)$ contains all nilpotent elements in $R$ (hence Köthe's conjecture holds for $R$) and $R/W(R)$ is a commutative reduced ring; (ii) $R$ is strongly bounded if $R/N_*(R)$ is commutative and $0\neq N_*(R)\subseteq Z(R)$, where $W(R)$ (resp., $N_*(R)$) is the Wedderburn (resp., prime) radical of $R$. We provide plenty of interesting examples that answer the questions raised in relation to the condition that $R/I$ is commutative and $I\subseteq Z(R)$. In addition, we study the structure of rings whose factor rings modulo nonzero proper ideals are commutative; such rings are called FC. We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Let <img src=image/13424240_02.gif> be a commutative ring with unity. The essential ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_03.gif>, is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are adjacent whenever <img src=image/13424240_06.gif> is an essential ideal. An essential ideal <img src=image/13424240_04.gif> of a ring <img src=image/13424240_02.gif> is an ideal <img src=image/13424240_04.gif> of <img src=image/13424240_02.gif> (<img src=image/13424240_07.gif>), having nonzero intersection with every other ideal of <img src=image/13424240_02.gif>. The set <img src=image/13424240_08.gif> contains all the maximal ideals of <img src=image/13424240_02.gif>. The Jacobson radical of <img src=image/13424240_02.gif>, <img src=image/13424240_09.gif>, is the set of intersection of all maximal ideals of <img src=image/13424240_02.gif>. The comaximal ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_11.gif>, is a simple graph with vertices as proper ideals of A not contained in <img src=image/13424240_09.gif> and the vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are associated with an edge whenever <img src=image/13424240_10.gif>. In this paper, we study the structural properties of the graph <img src=image/13424240_03.gif> by using the ring theoretic concepts. We obtain a characterization for <img src=image/13424240_03.gif> to be isomorphic to the comaximal ideal graph <img src=image/13424240_11.gif>. Moreover, we derive the structure theorem of <img src=image/13424240_12.gif> and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of <img src=image/13424240_12.gif> and determine the values of <img src=image/13424240_13.gif> for which <img src=image/13424240_12.gif> is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to <img src=image/13424240_12.gif> for some <img src=image/13424240_13.gif>.

On the normalized Laplacian spectrum of the comaximal graphs

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

On modules satisfying the descending chain condition on r-submodules

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we introduce the concept of r-Artinian modules which is a new generalization of Artinian modules. An R-module M is called an r-Artinian module if M satisfies the descending chain condition on r-submodules. Also, we call the ring R to be an r-Artinian ring if R is an r-Artinian R-module. We prove that an R-module M is an r-Artinian module if and only if its total quotient module is an Artinian module. In particular, we observe that r-Artinian modules generalize S-Artinian modules, for some particular multiplicatively closed subsets S of R. Also, we extend many properties of Artinian modules to r-Artinian modules. Finally, we use the idealization construction to give non-trivial examples of r-Artinian rings that are not Artinian.

On strongly dccr⋆ modules

In this paper, we introduce and study the concept of strongly dccr[Formula: see text] modules. Strongly dccr[Formula: see text] condition generalizes the class of Artinian modules and it is stronger than dccr[Formula: see text] condition. Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] a unital [Formula: see text]-module. A module [Formula: see text] is said to be strongly [Formula: see text] if for every submodule [Formula: see text] of [Formula: see text] and every sequence of elements [Formula: see text] of [Formula: see text], the descending chain of submodules [Formula: see text] of [Formula: see text] is stationary. We give many examples and properties of strongly dccr[Formula: see text]. Moreover, we characterize strongly dccr[Formula: see text] in terms of some known class of rings and modules, for instance in perfect rings, strongly special modules and principally cogenerately modules. Finally, we give a version of Union Theorem and Nakayama’s Lemma in light of strongly dccr[Formula: see text] concept.