Nonzero Ideals Research Articles

Adjacency matrix and eigenvalues of the zero divisor graph Γ(Zn)

Let R be a commutative ring with non-zero identity and Z^∗(R) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z^∗(R) are adjacent if and only if xy = 0. In this paper, the eigenvalues of Γ(Zn) for n = p^2q^2, where p and q are distinct primes, are investigated. Also, the girth, diameter, clique number and stability number of this graph are found.

On the cofiniteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity and $I$ be an ideal of $R$. Let $M$ and $N$ be two finitely generated $R$-modules. In this paper it is shown that, if ${\rm cd}(I,R)\leq 1$ and ${\rm pd_{R}}(M)<\infty$, then $H_{I}^{i}(M,N)$ are $I$-cofinite for each $i\geq 0$. Moreover, it is shown that if $\mbox{q}(I,R) \leq 1$ and ${\rm pd_{R}}(M)<\infty$, then the Bass numbers of generalized local cohomology modules $H_{I}^{i}(M,N)$ are finite for each $i\geq 0$. We also characterize the greatest integer $i$ such that $H_{I}^{i}(M,N)$ is not Artinian and $I$-cofinite.

S-Semiprime Submodules and S -Reduced Modules

This article introduces the concept of S -semiprime submodules which are a generalization of semiprime submodules and S -prime submodules. Let M be a nonzero unital R-module, where R is a commutative ring with a nonzero identity. Suppose that S is a multiplicatively closed subset of R . A submodule P of M is said to be an S -semiprime submodule if there exists a fixed s ∈ S , and whenever r n m ∈ P for some r ∈ R , m ∈ M , and n ∈ ℕ , then srm ∈ P . Also, M is said to be an S -reduced module if there exists (fixed) s ∈ S , and whenever r n m = 0 for some r ∈ R , m ∈ M , and n ∈ ℕ , then srm = 0 . In addition, to give many examples and characterizations of S -semiprime submodules and S -reduced modules, we characterize a certain class of semiprime submodules and reduced modules in terms of these concepts.

On S-Zariski topology

Let R be a commutative ring with nonzero identity and, be a multiplicatively closed subset. An ideal P of R with is called an S-prime ideal if there exists an (fixed) and whenver for then either or In this article, we construct a topology on the set SpecS (R) of all S-prime ideals of R which is generalization of prime spectrum of R. Also, we investigate the relations between algebraic properties of R and topological properties of SpecS (R) like compactness, connectedness and irreducibility.

Classification of rings with toroidal co-annihilating-ideal graphs

Let [Formula: see text] be a commutative ring with identity. The co-annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text]. In this paper, we study the planarity and genus of [Formula: see text]. In particular, we characterize all Artinian rings [Formula: see text] for which the genus of [Formula: see text] is zero or one.

Topological properties of the prime spectrum of a semimodule

The objective of this paper is to study topological properties of the prime spectrum of a unitary semimodule over a semiring with zero and nonzero identity. We define a top semimodule over a semiring, show that the prime spectrum is a T0-space, and prove that each irreducible closed subset of the prime spectrum has a generic point and that the prime spectrum is a compact space if the top semimodule is finitely generated. For a multiplication semimodule over a commutative semiring, we reveal the structure of the radical of a subsemimodule and prove that the multiplication semimodule is finitely generated iff the prime spectrum is a compact space, that in the prime spectrum, every basic open set is compact and the intersection of finitely many basic open sets is also compact, and that the prime spectrum is a spectral space if the multiplication semimodule is finitely generated.

Coloring of cozero-divisor graphs of commutative von Neumann regular rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with vertices in $$W^*(R)$$ , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in $$W^*(R)$$ are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.

Rings in which every 2-absorbing primary ideal is primary

Let R be a commutative ring with a nonzero identity. Badawi et al. (Bull Korean Math Soc 51(4):1163–1173, 2014) defined the concept of 2-absorbing primary ideal as follows: a proper ideal P of R is said to be a 2-absorbing primary ideal if whenever $$xyz\in P$$ for some $$x,y,z\in R,$$ then either $$xy\in P$$ or $$xz\in \sqrt{P}$$ or $$yz\in \sqrt{P}.$$ It is clear that every primary ideal is also a 2-absorbing primary ideal. The author in this paper is to study rings in which every 2-absorbing primary ideal is primary. A ring R is said to be $$2-ABP$$ -ring if every 2-absorbing primary ideal is primary.

A graph with respect to idempotents of a ring

Let [Formula: see text] be a ring with nonzero identity. The Idempotent graph of [Formula: see text], denoted by [Formula: see text], has its set of vertices equal to the set of all elements of [Formula: see text]; Distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an idempotent of [Formula: see text]. In this paper, we study some basic properties of [Formula: see text] such as connectedness, diameter and girth.

ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

In 2015, the second-named author introduced the dot product graph associated to a commutative ring $A$. Let $A$ be a commutative ring with nonzero identity, $1 \leq n < \infty$ be an integer, and $R = A \times A \times \cdots \times A$ ($n$ times). We recall that the total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^* = R\setminus \{(0, 0, \dots, 0)\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y = 0 \in A$ (where $x\cdot y$ denotes the normal dot product of $x$ and $y$). Let $Z(R)$ denote the set of all zero-divisors of $R$. Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZD(R)$ of $TD(R)$ with vertices $Z(R)^* = Z(R) \setminus \{(0, 0, \dots, 0)\}$. Let $U(R)$ denote the set of all units of $R$. Then the unit dot product graph of $R$ is the induced subgraph $UD(R)$ of $TD(R)$ with vertices $U(R)$. In this paper, we study the structure of $TD(R)$, $UD(R)$, and $ZD(R)$ when $A = Z_n$ or $A = GF(p^n)$, the finite field with $p^n$ elements, where $n \geq 2$ and $p$ is a prime positive integer.

Dhara–Rehman–Raza’s identities on left ideals of prime rings

The main aim of this paper is to generalize some results of Dhara et al. in (Miskolc Math Notes 16:769–779, 2015) to the context of nonzero left ideals. We start the paper with a result which shows that any square closed Lie ideal of a 2-torsion free prime ring contains a nonzero ideal.

The probability that ideals in a number ring are k-wise relatively r-prime

We say that [Formula: see text] nonzero ideals of algebraic integers in a fixed number ring are [Formula: see text]-wise relatively [Formula: see text]-prime if any [Formula: see text] of them are relatively [Formula: see text]-prime. In this paper, we provide an exact formula for the probability that [Formula: see text] nonzero ideals of algebraic integers in a fixed number ring are [Formula: see text]-wise relatively [Formula: see text]-prime.

Multiplicative (generalized)-derivations of prime rings that act as $n$-(anti)homomorphisms

Let R be a prime ring. In this note, we describe the possible forms of multiplicative (generalized)-derivations of R that act as n-homomorphism or n-antihomomorphism on nonzero ideals of R. Consequently, from the given results one can easily deduce the results of Gusić ([7]).

On the spectrum of the closed unit graphs

Let R be a finite commutative ring with non-zero identity. Let and be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by , is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if . If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit graph, denoted by . In this paper, we compute the adjacency spectrum of the graph . We utilize this result to show that if and only if and , where R and S are two arbitrary finite rings. Moreover, we determine when is a Ramanujan graph. We also deliver a necessary and sufficient condition for to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs.

Some results about ϕ-primal subsemimodules

Let [Formula: see text] be a commutative semiring with nonzero identity and [Formula: see text] an [Formula: see text]-semimodule. In this paper, we introduce the concept of [Formula: see text]-primal subsemimodule of [Formula: see text] that is a generalization of primal ideal of a commutative ring. Then we give some examples and properties of these subsemimodules. Also, some characterizations of [Formula: see text]-primal subsemimodules are presented.

Some Remarks on the Annihilator Graph of a Commutative Ring

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.

Domination in generalized unit and unitary Cayley graphs of finite rings

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that x + uy is a unit in R. Also, $$\overline{\Gamma}$$ denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as $$\overline{\Gamma}$$ and characterize all γ-sets in Γ and $$\overline{\Gamma}$$. Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of $$\overline{\Gamma}$$.

Strongly hollow elements of commutative rings

In this paper, the notion of completely strongly hollow ideals (respectively, strongly hollow elements) of a commutative ring as a generalization of strongly hollow ideals (respectively, local idempotents) is introduced and some related properties are investigated. Also, some characteristics of completely strongly hollow ideals and strongly hollow elements of some special classes of commutative rings are given. Then, we consider the decomposition of nonzero ideals of a ring into a product (respectively, (finite) sum) of completely strongly hollow ideals. Some other results are obtained too.

On the signless Laplacian and normalized Laplacian spectrum of the zero divisor graphs

Let R be a commutative ring with nonzero identity and let $$\Gamma (R)$$ denote the zero divisor graph of R. In this paper, we describe the signless Laplacian and normalized Laplacian spectrum of the zero divisor graph $$\Gamma (\mathbb {Z}_n)$$, and we determine these spectrums for some values of n. We also characterize the cases that 0 is a signless Laplacian eigenvalue of $$\Gamma (\mathbb {Z}_n)$$. Moreover, we find some bounds for some eigenvalues of the signless Laplacian and normalized Laplacian matrices of $$\Gamma (\mathbb {Z}_n)$$.

English

Let R be a commutative ring with nonzero identity, let I(R) be the set of all ideals of R and δ: I(R) → I(R) an expansion of ideals of R defined by I ↦ δ(I). We introduce the concept of (δ, 2)-primary ideals in commutative rings. A proper ideal I of R is called a (δ, 2)-primary ideal if whenever a, b ∈ R and ab ∈ I, then a2 ∈ I or b2 ∈ δ(I). Our purpose is to extend the concept of 2-ideals to (δ, 2)-primary ideals of commutative rings. Then we investigate the basic properties of (δ, 2)-primary ideals and also discuss the relations among (δ, δ-primary, δ-primary and 2-prime ideals.