Non-unit Elements Research Articles

On S-1-absorbing prime and weakly S-1-absorbing prime ideals

Let A be a commutative ring with identity and S be a multiplicative subset of A. In this paper, we introduce and study the notions of S-1-absorbing and weakly S-1-absorbing prime ideals as generalizations of the notion of prime ideals. We define a proper ideal I disjoint with S to be an S-1-absorbing (resp. a weakly S-1-absorbing) prime ideal if there exists s ∈ S such that for all nonunit elements a, b, c ∈ A such that abc ∈ I (resp. 0 ≠ abc ∈ I), we have sab ∈ I or sc ∈ I. Several properties and characterizations of S-1-absorbing prime and weakly S-1-absorbing prime ideals are given. Moreover, we study the transfer of the above properties to some constructions of rings such as trivial ring extensions and amalgamation of rings along an ideal.

Note on weakly 1-absorbing primary ideals

An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and 0 ? ab ? I, then a ? I or b ? ?I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ? R and 0 ? abc ? I, then ab ? I or c ? ?I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary.

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.

Comaximal graph of amalgamated algebras along an ideal

Let [Formula: see text] and [Formula: see text] be commutative rings with identity, [Formula: see text] be an ideal of [Formula: see text], and let [Formula: see text] be a ring homomorphism. The amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] denoted by [Formula: see text] was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of [Formula: see text] which are transferred to the comaximal graph of [Formula: see text], and also we study some algebraic properties of the ring [Formula: see text] by way of graph theory. The comaximal graph of [Formula: see text], [Formula: see text], was introduced by Sharma and Bhatwadekar in 1995. The vertices of [Formula: see text] are 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]. Let [Formula: see text] be the subgraph of [Formula: see text] generated by non-unit elements, and let [Formula: see text] be the Jacobson radical of [Formula: see text]. It is shown that the diameter of the graph [Formula: see text] is equal to the diameter of the graph [Formula: see text], and the girth of the graph [Formula: see text] is equal to the girth of the graph [Formula: see text], provided some special conditions.

Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.

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].

Classification of Rings with Toroidal and Projective Coannihilator Graph

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W S . The coannihilator graph of S , denoted by A G ′ S , is an undirected graph with vertex set W S ∗ (set of all nonzero nonunit elements of S ), and α ∼ β is an edge of A G ′ S ⇔ α ∉ α β S or β ∉ α β S , where δ S denotes the principal ideal generated by δ ∈ S . In this study, we first classify finite ring S , for which A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring S , for which A G ′ S is toroidal or projective.

Metric and Strong Metric Dimension in Cozero-Divisor Graphs

Let R be a commutative ring with non-zero identity and $$W^*(R)$$ be the set of all non-zero and non-unit elements of R. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with the vertex set $$W^*(R)$$ and two distinct vertices a and b are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, the metric dimension and strong metric dimension of $$\Gamma ^{\prime }(R)$$ are investigated. We compute the exact values of strong metric and metric dimension in cozero-divisor graphs of reduced rings. Moreover, the metric dimension in cozero-divisor graphs of non-reduced rings is discussed.

When Is a Puiseux Monoid Atomic?

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If M is a Puiseux monoid, then the question of whether each nonunit element of M can be written as a sum of irreducible elements (that is, M is atomic) is surprisingly difficult. For instance, although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are atomic, no general characterization of such monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux monoids. We provide characterizations of when M is finitely generated, factorial, half-factorial, other-half-factorial, Prüfer, seminormal, root-closed, and completely integrally closed. In addition to the atomic property, precise characterizations are also not known for when M satisfies the ACCP, is a BF-monoid, or is an FF-monoid; in each of these cases, we construct classes of Puiseux monoids satisfying these properties.

On S-1-absorbing prime submodules

Let [Formula: see text] be a commutative ring with non-zero identity, [Formula: see text] a multiplicatively closed subset of [Formula: see text] and [Formula: see text] a unital [Formula: see text]-module. In this paper, we introduce and study the concept of [Formula: see text]-1-absorbing prime submodules. A submodule [Formula: see text] of [Formula: see text] with [Formula: see text] is said to be [Formula: see text]-1-absorbing prime, if there exists an [Formula: see text] whenever [Formula: see text] for some non-unit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text]. Some examples, characterizations and properties of [Formula: see text]-1-absorbing prime submodules are given. Moreover, we give some characterizations of [Formula: see text]-1-absorbing prime submodules in multiplication modules.

On graded 1-absorbing prime ideals

Let G be a group with identity e and R be a G-graded commutative ring with unity 1. In this article, we introduce and study the concept of graded 1-absorbing prime ideals which is a generalization of graded prime ideals. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements $$x,y,z\in h(R)$$ such that $$xyz\in P$$ , then either $$xy\in P$$ or $$z\in P$$ .

Commutative rings with one-absorbing factorization

Let R be a commutative ring with nonzero identity. Yassine et al. defined the concept of 1-absorbing prime ideals as follows: a proper ideal I of R is said to be a 1-absorbing prime ideal if whenever for some nonunit elements then either or We use the concept of 1-absorbing prime ideals to study those commutative rings in which every proper ideal is a product of 1-absorbing prime ideals (we call them OAF-rings). Any OAF-ring has dimension at most one and local OAF-domains (D, M) are atomic such that M 2 is universal.

The elasticity of Puiseux monoids

Let M be an atomic monoid and let x be a non-unit element of M. The elasticity of x, denoted by ρ(x), is the ratio of its largest factorization length to its shortest factorization length, and it measures how far x is from having all its factorizations of the same length. The elasticity ρ(M) of M is the supremum of the elasticities of all non-unit elements of M. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of ℚ≥0). We characterize, in terms of the atoms, which Puiseux monoids M have finite elasticity, giving a formula for ρ(M) in this case. We also classify when ρ(M) is achieved by an element of M. When M is a primary Puiseux monoid (that is, a Puiseux monoid whose atoms have prime denominator), we describe the topology of the set of elasticities of M, including a characterization of when M is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most 2).

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.

Factorizations in upper triangular matrices over information semialgebras

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative monoids (and, in particular, for multiplicative monoids of integral domains), FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n≥2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative monoid Tn(S)• consisting of n×n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces Tn(S)• by its submonoid Un(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids Tn(S)• and Un(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.

On 1-absorbing prime ideals of commutative rings

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

Designing S-boxes triplet over a finite chain ring and its application in RGB image encryption

In this article, we have developed an encryption technique to encrypt any kind of digital information. The main work is to construct the component of the block ciphers namely the substitution boxes (S-boxes) over an algebraic structure of finite chain ring; and then use these S-boxes in image encryption applications. The present formation is based on the finite commutative chain ring R9 = ℤ2 + uℤ2 + … + uk − 1ℤ2 (module over itself) which is exactly twice of 256. The multiplicative group of unit elements of R9 precisely has 256 elements and the set of all nonunit elements in R9 forms a submodule of module R9 consisting of 256 elements. By using this point we initiate a new 8 × 8 S-box triplet generation technique which addresses the group of units of R9 and the submodule of R9. This new construction technique of S-boxes ensures random values in the area of the initial domain of transformation. The proposed S-boxes have been examined by algebraic, statistical and texture analyses. A comparison of the expected and existing S-boxes reveals that the proposed S-boxes are comparatively better and can be used in the well known ciphers. Another goal of this work is to suggest an encryption technique for colored (RGB) images based on permutation keys and triplet of newly generated S-boxes. The outcomes of the security, statistical and the differential analyses have proved that our scheme is better for the image encryption.

Co-maximal Graph, its Planarity and Domination Number

Let R be a finite commutative ring with identity. The co-maximal graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also, Γ2(R) is the subgraph of Γ(R) induced by non-unit elements and [Formula: see text] where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which [Formula: see text], Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.

The coloring of the cozero-divisor graph of a commutative ring

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and 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]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.