Non-unit Elements Research Articles

A generalization of the cozero-divisor graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be an ideal of [Formula: see text]. The zero-divisor graph of [Formula: see text] with respect to [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the set [Formula: see text] with distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The cozero-divisor graph [Formula: see text] of [Formula: see text] is the graph whose vertices are precisely the non-zero, non-unit 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] and [Formula: see text]. In this paper, we introduced and investigated a new generalization of the cozero-divisor graph [Formula: see text] of [Formula: see text] denoted by [Formula: see text]. In fact, [Formula: see text] is a dual notion of [Formula: see text].

On ϕ-S-1-absorbing δ-primary superideals over commutative super-rings

Let [Formula: see text] be a commutative super-ring with unity [Formula: see text] and let [Formula: see text] be the set of all superideals of [Formula: see text]. Let [Formula: see text] be a reduction function of superideals of [Formula: see text] and let [Formula: see text] be an expansion function of superideals of [Formula: see text]. We recall that a proper superideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-1-absorbing [Formula: see text]-primary superideal of [Formula: see text], if whenever [Formula: see text] for some nonunit elements [Formula: see text], then [Formula: see text] or [Formula: see text]. In this paper, we introduce a new class of superideals that is a generalization to the class of [Formula: see text]-1-absorbing [Formula: see text]-primary superideals. Let [Formula: see text] be a multiplicative subset of the subset [Formula: see text] of the homogeneous elements such that [Formula: see text] and let [Formula: see text] be a proper superideal of [Formula: see text] with [Formula: see text], then [Formula: see text] is called a [Formula: see text]-[Formula: see text]-1-absorbing [Formula: see text]-primary superideal of [Formula: see text] associated to [Formula: see text] if whenever [Formula: see text] for some nonunit elements [Formula: see text], then [Formula: see text] or [Formula: see text]. In this paper, we have presented a range of different examples, properties and characterizations of this new class of superideals.

On Ф-s-1-absorbing δ-primary ideals of commutative rings

Let R be a commutative ring with unity (1 ? 0) and let J(R) be the set of all ideals of R. Let ? : J(R) ? J(R) ? {?} be a reduction function of ideals of R and let ? : J(R) ? J(R) be an expansion function of ideals of R. We recall that a proper ideal I of R is called a ? -1-absorbing ?-primary ideal of R, if whenever abc ? I ? ? (I) for some nonunit elements a, b, c ? R, then ab ? I or c ? ?(I). In this paper, we introduce a new class of ideals that is a generalization to the class of ?-1-absorbing ?-primary ideals. Let S be a multiplicative subset of R such that 1 ? S and let I be a proper ideal of R with S ? I = ?, then I is called a ?-S-1-absorbing ?-primary ideal of R associated to s ? S, if whenever abc ? I? ?(I) for some nonunit elements a, b, c ? R, then sab ? I or sc ? ?(I). In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.

Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛

Abstract Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

The Comaximal Graphs of Noncommutative Rings

For a ring [Formula: see text] (not necessarily commutative) with identity, the comaximal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all the nonunit 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 consider a subgraph [Formula: see text] of [Formula: see text] induced by [Formula: see text], where [Formula: see text] is the set of all left-invertible elements of [Formula: see text]. We characterize those rings [Formula: see text] for which [Formula: see text] is a complete graph or a star graph, where [Formula: see text] is the Jacobson radical of [Formula: see text]. We investigate the clique number and the chromatic number of the graph [Formula: see text], and we prove that if every left ideal of [Formula: see text] is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of [Formula: see text]. We also investigate the properties of [Formula: see text] when [Formula: see text] is a split graph.

A study of upper ideal relation graphs of rings

Let R be a ring with unity. The upper ideal relation graph of the ring R is the simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of . We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph is planar and outerplanar, respectively. For a non-local commutative ring , where Ri is a local ring with maximal ideal and , we prove that the graph is perfect if and only if and each is a principal ideal. We also discuss all the finite rings R such that the graph is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of , when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of . We classify all the values of n for which the graph is Hamiltonian.

Atomic semigroup rings and the ascending chain condition on principal ideals

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by A. Grams in the seventies with a neat counterexample. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams’ first construction. In the first part of this paper, we generalize Grams’ construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct a new class of atomic semigroup rings without the ACCP.

Amalgamated algebras along an ideal defined by 1-absorbing-like conditions

Let [Formula: see text] be a commutative ring with nonzero identity. A proper ideal [Formula: see text] of [Formula: see text] is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements [Formula: see text] with [Formula: see text] then [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text]). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] (denoted by [Formula: see text]), introduced and studied by D’Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties.

Pseudo subalgebras and pseudo filters in pseudo BE-algebras

<abstract><p>As a generalization of BE-algebras, the pseudo BE-algebra was introduced by Borzooei et al., and the notions of pseudo subalgebras and pseudo filters in pseudo BE-algebras were defined, and some related properties were investigated. In order to further study pseudo subalgebras and pseudo filters in pseudo BE-algebras, concepts of pseudo atom, atomic pseudo BE-algebra, and atomic pseudo filter are introduced and related studies are conducted. The conditions under which pseudo-filters can be created using a nonempty set, and the conditions under which non-unit elements can be pseudo-atoms are explored. Characterization of atomic pseudo BE-algebra is discussed, and conditions are provided under which pseudo subalgebra can be pseudo filters. The relationship between a set of pseudo atoms and a pseudo subalgebra is considered, and the conditions under which a pseudo filter can be atomic are found.</p></abstract>

On Weakly 1-Absorbing Primary Ideals of Commutative Semirings

Let $R$ be a commutative semiring with $ 1 \neq0$. In this paper, we study the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing ideal over commutative semirings . A proper ideal $I$ of a semiring $R$ is called a weakly 1-absorbing primary ideal if whenever nonunit elements $a,b,c \in R$ and $0 \neq abc \in I$, then $ab \in I $, or $c \in \sqrt{I}$. A number of results concerning weakly 1-absorbing primary ideals and examples of weakly 1-absorbing primary ideals are given. An ideal is called a subtractive ideal $I$ of a semiring $R$ is an ideal such that if $ x,x+y\in I$, then $ y\in I$. Subtractive ideals or k-ideals are helpful in proving in many results related to ideal theory over semirings.

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.

The Krull dimension-dependent elements of a Noetherian commutative ring

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let [Formula: see text] be non-unit elements of a commutative ring [Formula: see text]. [Formula: see text] are called Krull dimension-dependent elements, whenever [Formula: see text] We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension.

A Multiplicative Dual Nil Q-Clean Rings

In this paper our goal to thoroughly determine the rings in which each non-unit element is a product of a nilpotent and a quasi-idempotent.

On the cozero-divisor graphs associated to rings

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if and In this paper, first we study the Laplacian spectrum of We show that the graph is Laplacian integral. Further, we obtain the Laplacian spectrum of for where and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of we characterized the values of n for which the Laplacian spectral radius is equal to the order of Moreover, the values of n for which the algebraic connectivity and vertex connectivity of coincide are also described. At the final part of this paper, we obtain the Wiener index of for arbitrary n.

On UQ rings

In this paper, we introduce [Formula: see text] rings. A ring [Formula: see text] is called [Formula: see text] if every non-unit element of [Formula: see text] can be represented as a product of a unit and a quasiregular element. We provide various properties of [Formula: see text] rings along with its characterizations. We give a new characterization of [Formula: see text]-good rings, and it turns out that [Formula: see text]-good rings are precisely the rings in which every element is a product of a unit and a quasiregular element. We discuss various extensions of [Formula: see text] rings such as Morita contexts, generalized matrix rings, formal matrix rings, group rings, etc.

The adjacency spectrum and metric dimension of an induced subgraph of comaximal graph of ℤn

Let R be a commutative ring with unity, and let [Formula: see text] denote the comaximal graph of R. The comaximal graph [Formula: see text] has vertex set as R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. Let [Formula: see text] denote the induced subgraph of [Formula: see text] on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. In this paper, we study the graphical structure as well the adjacency spectrum of [Formula: see text], where [Formula: see text] is a non-prime positive integer, and [Formula: see text] is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of [Formula: see text] are [Formula: see text] with multiplicity [Formula: see text], and remaining eigenvalues are contained in the spectrum of a symmetric [Formula: see text] matrix. We further calculate the rank and nullity of [Formula: see text]. We also determine all the eigenvalues of [Formula: see text] whenever [Formula: see text] is a bipartite graph. Finally, apart from determining certain structural properties of [Formula: see text], we conclude the paper by determining the metric dimension of [Formula: see text].

On Weakly 1-Absorbing Primary Ideals of Commutative Rings

Let [Formula: see text] be a commutative ring with [Formula: see text]. We introduce the concept of weakly 1-absorbing primary ideal, which is a generalization of 1-absorbing primary ideal. A proper ideal [Formula: see text] of [Formula: see text] is said to be weakly 1-absorbing primary if whenever nonunit elements [Formula: see text] and [Formula: see text], we have [Formula: see text] or [Formula: see text]. A number of results concerning weakly 1-absorbing primary ideals are given, as well as examples of weakly 1-absorbing primary ideals. Furthermore, we give a corrected version of a result on 1-absorbing primary ideals of commutative rings.

Idempotent Divisor Graph of Commutative Ring

This work aims to introduce and to study a new kind of divisor graph which is called idempotent divisor graph, and it is denoted by . Two non-zero distinct vertices v1 and v2 are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of , as well as it’s connection with . We also study planarity of this graph.

Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.