Nonzero Ideals Research Articles

On nonnil S-SFT rings

Let [Formula: see text] be a commutative ring with nonzero identity element and S a multiplicative subset of [Formula: see text]. In this paper, we introduce and investigate the notion of nonnil S-SFT rings. The ring [Formula: see text] is said to be nonnil S-SFT, if for each nonnil-ideal [Formula: see text] of [Formula: see text], there exist [Formula: see text], [Formula: see text] and a finitely generated ideal [Formula: see text] such that [Formula: see text] for all [Formula: see text], in that case, the ideal [Formula: see text] is called S-SFT. It is shown that the ring [Formula: see text] is nonnil S-SFT ring if and only if each nonnil-prime ideal (disjoint with S) is S-SFT. The transfert of the nonnil S-SFT concept to flat overrings and trivial extension is investigated. We give several characterizations of nonnil S-SFT rings. Also, we give a characterization of the nonnil-SFT variant in terms of the nonnil S-SFT variant.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Right ideal decompositions of nearrings with faithful 2-step modules

When N is a left nearring with identity satisfying the descending chain condition on right ideals N can be written as a direct sum of nonzero right ideals that cannot be further decomposed into direct sums of nonzero right ideals. If N is a ring the nature of this decomposition is well understood. A nearring module of N possessing a unique proper nonzero N-subgroup is called a 2-step module. Here we initiate the study of the nature of the previously described decompositions when N is not a ring by considering the situation where N has a faithful 2-step module.

On rings whose prime ideal sum graphs are line graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

On Graded 2-n-Submodules of Graded Modules Over Graded Commutative Rings

In this article, all rings are commutative with a nonzero identity. Let G be a group with identity e, R be a G-graded commutative ring, and M be a graded R-module. In 2019, the concept of graded n-ideals was introduced and studied by Al-Zoubi, Al-Turman, and Celikel. A proper graded ideal I of R is said to be a graded n-ideal of R if whenever r,s∈h(R) with rs∈I and r∉Gr(0), then s∈I. In 2023, the notion of graded n-ideals was recently extended to graded n-submodules by Al-Azaizeh and Al-Zoubi. A proper graded submodule N of a graded R-module M is said to be a graded n-submodule if whenever t∈h(R), m∈h(R) with tm∈N and t∉Gr(Ann_R (M)), then m∈N. In this study, we introduce the concept of graded 2-n-submodules of graded modules over graded commutative rings generalizing the concept of graded n-submodules. We investigate some characterizations of graded 2-n-submodules and investigate the behavior of this structure under graded homomorphism and graded localization. A proper graded submodule U of M is said to be a graded 2-n-submodule if whenever r,s∈h(R), m∈(M) and rsm∈U, then rs∈Gr(Ann_R (M)) or rm∈U or tm∈U.

On the unit-Jacobson graph

In this paper, we introduce the unit-Jacobson graph, which is defined by the unit elements and the elements of the Jacobson radical of a commutative ring [Formula: see text] with nonzero identity. We give relationships between this new graph concept and some special rings such as [Formula: see text]-clean rings, [Formula: see text]-rings, local rings, and cartesian rings. Moreover, we investigate the concepts of the dominating set, diameter, and girth on the unit-Jacobson graph.

Subfield codes of CD-codes over [formula omitted]

A non-zero F-linear map from a finite-dimensional commutative F-algebra to the field F is called an F-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an F2-valued trace of the F2-algebra R2:=F2[x]/〈x3−x〉 to study binary subfield code CD(2) of CD:={(x⋅d)d∈D:x∈R2m} for each defining set D derived from a certain simplicial complex. For m∈N and X⊆{1,2,…,m}, define ΔX:={v∈F2m:Supp(v)⊆X} and D:=(1+u2)D1+u2D2+(u+u2)D3, a subset of R2m, where u=x+〈x3−x〉,D1∈{ΔL,ΔLc},D2∈{ΔM,ΔMc} and D3∈{ΔN,ΔNc}, for L,M,N⊆{1,2,…,m}. The parameters and the Hamming weight distribution of the binary subfield code CD(2) of CD are determined for each D. These binary subfield codes are minimal under certain mild conditions on the cardinalities of L,M and N. Moreover, most of these codes are distance-optimal. Consequently, we obtain a few infinite families of minimal, self-orthogonal and distance-optimal binary linear codes that are either 2-weight or 4-weight. It is worth mentioning that we have obtained several new distance-optimal binary linear codes.

Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a , b ∈ R \\ { 0 } . The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I ⊆ RaR of R or aJ = 0 for some nonzero ideal J ⊆ RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I ⊆ RaR and J ⊆ RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.

$\delta (0)$-Ideals of Commutative Rings

Let $R$ be a commutative ring with nonzero identity, let $\I (R)$ be the set of all ideals of $R$ and $\delta : \I (R)\rightarrow\I (R) $ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J \subseteq I$, we have $L \subseteq\delta(L)$ and $\delta(J)\subseteq\delta(I)$. In this paper, we present the concept of $\dt$-ideals in commutative rings. A proper ideal $I$ of $R$ is called a $\dt$-ideal if whenever $a$, $b$ $\in R$ with $ab\in I$ and $a\notin \delta (0)$, we have $b\in I$. Our purpose is to extend the concept of $n$-ideals to $\dt$-ideals of commutative rings. Then we investigate the basic properties of $\dt$-ideals and also, we give many examples about $\dt$-ideals.

Graded amalgamated algebras along an ideal defined by graded 1-absorbing-like conditions

Let [Formula: see text] be a group with identity and [Formula: see text] be a [Formula: see text]-graded commutative ring with nonzero identity. A proper graded ideal [Formula: see text] of [Formula: see text] is called a graded 1-absorbing prime ideal (respectively, graded 1-absorbing primary ideal) if whenever nonunit homogeneous elements [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text], where [Formula: see text] is the graded radical of [Formula: see text]). The purpose of this paper is to study the transfer of some graded 1-absorbing-like properties to the graded amalgamated algebra along an ideal (denoted by [Formula: see text]). Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties.

Extended total graph associated to finite commutative rings

UDC 512.5 For a commutative ring R with nonzero identity 1 ≠ 0 , let Z ( R ) denote the set of zero divisors. The total graph of R denoted by T Γ ( R ) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z ( R ) . In this paper, we define an extension of the total graph denoted by T ( Γ e ( R ) ) with vertex set as Z ( R ) , and two distinct vertices x and y are adjacent if and only if x + y ∈ Z * ( R ) , where Z * ( R ) is the set of nonzero zero divisors of R . Our main aim is to characterize the finite commutative rings whose T ( Γ e ( R ) ) has clique numbers 1,2 , and 3 . In addition, we characterize finite commutative nonlocal rings R for which the corresponding graph T ( Γ e ( R ) ) has the clique number 4.

The dimension graph of a commutative ring

This paper introduces a simple graph associated to a commutative ring. Nonzero ideals I and J of a commutative ring R are called Krull dimension-dependent whenever dim R / ( I + J ) = min { dim R / I , dim R / J } . Based on this, we introduce the dimension graph of R. We study and investigate this graph, and we determine all rings with disconnected dimension graph. Moreover, we introduce strongly Krull dimension-dependent ideals, and we examine these ideals using a subgraph of the dimension graph.

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Let R be a commutative ring with unity. The prime ideal sum graph PIS ( R ) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that PIS ( R ) is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.

Quantitative upper bounds related to an isogeny criterion for elliptic curves

AbstractFor and elliptic curves defined over a number field , without complex multiplication, we consider the function counting nonzero prime ideals of the ring of integers of , of good reduction for and , of norm at most , and for which the Frobenius fields and are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that and are not potentially isogenous if and only if , we investigate the growth in of . We prove that if and are not potentially isogenous, then there exist positive constants , , and such that the following bounds hold: (i) ; (ii) under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) under GRH, Artin's Holomorphy Conjecture for the Artin ‐functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin ‐functions of number field extensions.

On graded pseudo 2-prime ideals

In this paper, we study graded pseudo 2-prime ideals of graded commutative rings with nonzero identities. Let G be a commutative additive monoid with an identity element 0, and R=⊕g∈G Rg be a commutative graded ring with a nonzero identity element. A proper graded ideal I of R is said to be a graded pseudo 2-prime ideal if whenever ab ∈ I for some homogeneous elements a,b ∈ R, then a²ⁿ ∈ Iⁿ or b²ⁿ ∈ Iⁿ for some n ∈ ℕ. Besides giving many properties of graded pseudo 2-prime ideals, we characterize graded almost valuation domains in terms of our new concept.

Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

Properties of the ideal-intersection graph of the ring Zn

In this paper we study properties of the ideal-intersection graph of the ring Zn. The graph of ideal intersections is a simple graph in which the vertices are non-zero ideals of the ring, and two vertices (ideals) are adjacent if their intersection is also a non-zero ideal of the ring. These graphs can be referred to as the intersection scheme of equivalence classes (See: Laxman Saha, Mithun Basak Kalishankar Tiwary “Metric dimension of ideal-intersection graph of the ring Zn” [1] ).In this article we prove that the triameter of graph is equal to six or less than six. We also describe maximal clique of the ideal-intersection graph of the ring Zn. We prove that the chromatic number of this graph is equal to the sum of the number of elements in the zero equivalence class and the class with the largest number of element. In addition, we demonstrate that eccentricity is equal to 1 or it is equal to 2. And in the end we describe the central vertices in the ideal-intersection graph of the ring Zn.

CONSTRUCT THE TRIPLE ZERO GRAPH OF RING Z_n USING PYTHON

Let be a commutative ring with nonzero identity and there exists such that , , , denotes the set of all triple zero elements of . The triple zero graph of , denoted by , is an undirected graph with vertex set where two distinct vertices and are adjacent if and only if , and there exists a nonzero element of such that , , and . Python is a programming language with simple and easy-to-learn code that can be used to solve problems in algebra and graphs. In this paper, we construct the triple zero graph of ring using Python. Based on the output of the program, several properties of are obtained, such as if and , then is a planar graph, if with is prime numbers, then is a complete graph , and if with is prime numbers and , then is a connected graph.

Formula omitted]-valued trace of a finite-dimensional commutative [formula omitted]-algebra

A non-zero F-valued F-linear map on a finite-dimensional commutative F-algebra is called an F-valued trace if its kernel contains no non-zero ideals. However, given an F-algebra, such a map may not always exist. We find an infinite class of finite-dimensional commutative F-algebras which admit an F-valued trace. In fact, in these cases, we explicitly construct a trace map. An F-valued trace on a finite-dimensional commutative F-algebra induces a non-degenerate bilinear form on the F-algebra which may be theoretically and computationally helpful. In this article, we suggest a couple of applications of an F-valued trace map of an F-algebra to algebraic coding theory.

Some Properties of S-Semiannihilator Small Submodules and S-Small Submodules with respect to a Submodule

Let R be a commutative ring with nonzero identity, S⊆R be a multiplicatively closed subset of R, and M be a unital R-module. In this article, we introduce the concepts of S-semiannihilator small submodules and S-T-small submodules as generalizations of S-small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.