Nonzero Ideals Research Articles

Coidempotent graph of a ring

Let [Formula: see text] be a ring with nonzero identity. The coidempotent graph of [Formula: see text], denoted it 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] are adjacent in [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is an idempotent of [Formula: see text]. In this paper, we study some basic properties of [Formula: see text] and find some results about the ring-theoretic properties of [Formula: see text] and graph-theoretic properties of [Formula: see text].

On weakly 1-absorbing primary submodules

In this paper, we introduce weakly 1-absorbing primary submodules of modules over commutative rings. Let [Formula: see text] be a commutative ring with a nonzero identity and [Formula: see text] be a nonzero unital module. A proper submodule [Formula: see text] of [Formula: see text] is said to be a weakly 1-absorbing primary submodule if whenever [Formula: see text] for some nonunit elements [Formula: see text] and [Formula: see text] then [Formula: see text] or [Formula: see text]-[Formula: see text] where [Formula: see text]-[Formula: see text] is the prime radical of [Formula: see text] Many properties and characterizations of weakly 1-absorbing primary submodules are given. We also give the relations between weakly 1-absorbing primary submodules and other classical submodules such as weakly prime, weakly primary, weakly 2-absorbing primary submodules. Also, we use them to characterize simple modules.

On (1,r)-ideals of commutative rings

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of [Formula: see text]-ideals. A proper ideal [Formula: see text] of [Formula: see text] is said to be a [Formula: see text]-ideal if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some basic properties of [Formula: see text]-ideals are studied. For instance, we give a method to construct [Formula: see text]-ideals that are not [Formula: see text]-ideals. Among other things, it is shown that if [Formula: see text] admits a [Formula: see text]-ideal that is not an [Formula: see text]-ideal, then [Formula: see text] is a local ring. Also, we provide a necessary and sufficient condition in term of [Formula: see text]-ideals for a ring to be a total quotient ring and we determinate the [Formula: see text]-ideals of a chained ring. Finally, we give an idea about some [Formula: see text]-ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of [Formula: see text]-ideals.

Distributive invariant centrally essential rings

In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings strongly extends the class of commutative rings. For such rings, a number of recent papers contain positive answers to some important questions from ring theory that previously had positive answers for commutative rings and negative answers in the general case. This work is devoted to a similar topic. A familiar description of right Noetherian, right distributive centrally essential rings is generalized on a larger class of rings. Let A be a ring with prime radical P(A). It is proved that A is a right distributive, right invariant centrally essential ring and P(A) is a finitely generated right ideal such that the factor-ring [Formula: see text] does non contain an infinite direct sum of nonzero ideals if and only if [Formula: see text], where every ring [Formula: see text] is either a commutative Prüfer domain or an Artinian uniserial ring. The studies of Tuganbaev are supported by Russian scientific foundation project 22-11-00052.

A characterization of Krull domains in terms of their factor rings

ABSTRACT Let D be an integral domain, w be the so-called w-operation on D, and be the set of nonzero finitely generated ideals J of D with . Let I be a w-ideal of D, D/I be the factor ring of D modulo I, be the injective envelope of D/I, and . Recently Zhou, Kim, and Hu showed that is a commutative ring with identity containing D/I. The Krull domains are a w-analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w-invertible. It is well known that D is a Dedekind domain if and only if D/aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits , if and only if D/I is a zero-dimensional semilocal PIR for all nonzero ideals I of D. In this paper, we prove that D is a Krull domain if and only if is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if is a zero-dimensional semilocal PIR for any w-ideal I of D. We also study when is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w-ideals I of D.

Nearly Quasi Primary-2-Absorbing Submodules

Let be a nonzero unital left -module and be a commutative ring with nonzero identity. As a generalization of 2-Absorbing submodules, we provide the idea of Nearly Quasi Primary-2-Absorbing submodules in this article. As a proper submodule of is called the Nearly Quasi Primary-2-Absorbing submodule of , if whenever for , , implies that either or or . Several properties, characterizations and examples concerning this new notion are given.

ϕ - δ -Primary Hyperideals in Krasner Hyperrings

In this paper, we study commutative Krasner hyperrings with nonzero identity. ϕ -prime, ϕ -primary and ϕ - δ -primary hyperideals are introduced. The concept of δ -primary hyperideals is extended to ϕ - δ -primary hyperideals. Some characterizations of hyperideals are provided to classify them. The relation between ϕ - δ -primary hyperideals and other hyperideals is discussed.

On the signless Laplacian spectrum of the comaximal graphs

Assume that [Formula: see text] is a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertex set consists of all elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate the signless Laplacian spectrum of the comaximal graph of the ring [Formula: see text].

Graded weakly 1-absorbing prime ideals

In this paper, we introduce and study graded weakly 1-absorbing prime ideals in graded commutative rings. Let \(G\) be a group and \(R\) be a \(G\)-graded commutative ring with a nonzero identity \(1\neq0\). A proper graded ideal \(P\) of \(R\) is called a graded weakly 1-absorbing prime ideal if for each nonunits \(x,y,z\in h(R)\) with \(0\neq xyz\in P\), then either \(xy\in P\) or \(z\in P\). We give many properties and characterizations of graded weakly 1-absorbing prime ideals. Moreover, we investigate weakly 1-absorbing prime ideals under homomorphism, in factor ring, in rings of fractions, in idealization.

On the extended graph associated with the set of all non-zero annihilating ideals of a commutative ring

Let Rbe a commutative ring with non-zero identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r∈R\{0} such that Ir=(0). Let A(R) denote the set of all annihilating ideals of R and A (R)*=A(R)\{0}. In this article, we introduce a new graph associated with R denoted by H(R) whose vertex set is A(R)* and two distinct vertices I, J are adjacent in this graph if and only if IJ=(0) or I+J ∈ A(R). The aim of this article is to study the interplay between the ring-theoretic properties of a ring R and the graph-theoretic properties of H(R). For such a ring R, we prove that H(R) is connected and find its diameter. Moreover, we determine girth of H(R). Furthermore, we provide some sufficient conditions under which H(R) is a complete graph.

Graded S-1-absorbing prime submodules in graded multiplication modules

Let $G$ be a group with identity $e$. Let $R$ be a commutative $G$-graded ring with non-zero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ a graded $R$-module. In this article, we introduce and study the concept of graded $S$-1-absorbing prime submodules. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-1-absorbing prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then either $s_{g}a_{h}b_{h'}\in (N:_{R}M)$ or $s_{g}m_{k}\in N$ for all non-unit elements $a_{h},b_{h'}\in h(R)$ and all $m_{k}\in h(M)$. Some examples, characterizations and properties of graded $S$-1-absorbing prime submodules are given. Moreover, we give some characterizations of graded $S$-1-absorbing prime submodules in graded multiplicative modules.

A large class of graphs with a small subclass of Cohen–Macaulay members

Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we determine when these graphs are well-covered, and then, by applying this result, we characterize the unit graphs whose edge rings are Cohen–Macaulay (Gorenstein). This characterization gives us a large class of non-Cohen–Macaulay graphs.

Notes on 1-Absorbing Prime Ideals

Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyz ∈ I for some nonunits x, y, z ∈ R, then xy ∈ I or z ∈ I. It is well known that prime ideal ⇒ 1-absorbing prime ideal ⇒ primary ideal ⇒ semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings.

On Wiener Index of Unit Graph Associated with a Commutative Ring

Let [Formula: see text] be a ring with non-zero identity. The unit graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with all the elements of [Formula: see text] as vertices and where distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a unit of [Formula: see text]. In this paper, we investigate the Wiener index and hyper-Wiener index of [Formula: see text] and explicitly determine their values.

On the Aα spectrum of the zero-divisor graphs

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertex set consists of all elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text]. Specially, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct prime numbers and [Formula: see text]. Moreover, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text].

Prime ideal sum graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition [Formula: see text] is complete. Moreover, the diameter and the girth of [Formula: see text] are studied.

Unitary Cayley graphs whose Roman domination numbers are at most four

Let R be a finite commutative ring with nonzero identity. The unitary Cayley graph of R is the graph obtained by letting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x – y is a unit element of R. In this paper, we characterize all unitary Cayley graphs with Roman domination number at most four.

The book thickness of nilpotent graphs

We find the exact book thickness of all nilpotent graphs associated to commutative rings with nonzero identity whose genus is at most one. Also, we investigate the book thickness of nilpotent graphs from local rings and some product rings.

On φ-β-Absorbing Submodules

In this paper, we extend the concept of β-absorbing submodules to φ-β-absorbingsubmodules over a commutative ring with nonzero identity which is a generalization of 2-absorbing submodules. Let S(M) be the set of all submodules of M and φ : S(M) → S(M)∪{∅} be a function. A proper submodule P of M is called a φ-β-absorbing submodule, if for each r, s ∈ R and m ∈ M with rsm ∈ P\φ(P), then rs + rs ∈ (P : M) or r(m + m) ∈ P or s(m + m) ∈ P. Some of the properties and characterizations of φ-β-absorbing submodules are investigated.

Riesz and pre-Riesz monoids

Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all $$x,y_{1},y_{2}\ge 0$$ in M, $$x\le y_{1}+y_{2}\Rightarrow x=x_{1}+x_{2}$$ where $$0\le x_{i}\le y_{i}$$ . We explore the necessary and sufficient conditions under which a Riesz monoid M with $$M^{+}=\{x\ge 0\mid x\in M\}=M$$ generates a Riesz group and indicate some applications. We call a directed p.o. monoid M $$\Pi $$ -pre-Riesz if $$ M^{+}=M$$ and for all $$x_{1},x_{2}, \dots ,x_{n}\in M$$ , $${{\,\mathrm{glb}\,}}(x_{1},x_{2},\dots ,x_{n})=0$$ or there is $$r\in \Pi $$ such that $$0<r\le x_{1},x_{2},\dots ,x_{n},$$ for some subset $$\Pi $$ of M. We explore examples of $$\Pi $$ -pre-Riesz monoids of $$*$$ -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and $$\Pi $$ the set of invertible ideals, M is $$\Pi $$ -pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain.