The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R and finite commutative local ring Ri (1 < i < m), where R Ri and K2(R) = K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.     

On the Generalized Strongly Nil-Clean Property of Matrix Rings

Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n × n matrix A over R is expressible as a sum A =E1 +…+ Es + N of (commuting) idempotent matrices Ei and a nilpotent matrix N.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

Cut Vertices and Degree One Vertices of Zero-Divisor Graphs

This article continues to examine cut vertices in the zero-divisor graphs of commutative rings with 1. The main result is that, with only seven known exceptions, the zero-divisor graph of a commutative ring has a cut vertex if and only if the graph has a degree one vertex. This naturally leads to an examination of the degree one vertices of zero-divisor graphs.     

Classification of Rings with Genus One Zero-Divisor Graphs

This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has genus one.     

Zero-divisor Labelings of Graphs

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph.     

半质环的交换性条件

本文对满足可变恒等式的半质环在某种有界条件下给出了一个判断环Ｒ交换性的简便准则，使文献［２－２７］中所有相应结果均成为其直接推论．此外，对不限有界的情况，也得到较为广泛的结论．     

Zero-divisor graphs of idealizations

We consider zero-divisor graphs of idealizations of commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the zero-divisor graph of a ring when extending to idealizations of the ring.     

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

Let R be a commutative ring,I an ideal of R and k ≥ 2 a fixed integer.The ideal-based k-zero-divisor hypergraph HkI(R) of R has vertex set ZI(R,k),the set of all ideal-based k-zero-divisors of R,and for distinct elements x1,x2,…,xk in ZI(R,k),the set {x1,x2,…,xk} is an edge in HkI(R) if and only if x1x2…xk ∈ I and the product of the elements of any (k-1)-subset of {x1,x2,…,xk} is not in I.In this paper,we show that H3I(R) is connected with diameter at most 4 provided that x2 (∈) I for all ideal-based 3-zero-divisor hypergraphs.Moreover,we find the chromatic number of H3 (R) when R is a product of finite fields.Finally,we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H3I (R) planar.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

Zero-Divisor Graphs of Matrices Over Commutative Rings

We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M _n (R).     

Weakly central-vertex complete graphs with applications to commutative rings

The zero-divisor graph of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 1. This characterization is then applied to the class of zero-divisor graphs of commutative rings. For finite commutative rings whose zero-divisor graphs are not isomorphic to that of Z₄[X]/(X²), it is shown that weak central-vertex completeness is equivalent to the annihilator condition. Furthermore, a schema for describing zero-divisor graphs of radius 1 is provided.     

The embedding of line graphs associated to the zero-divisor graphs of commutative rings

Let R be a commutative ring with identity and denote Γ(R) for its zero-divisor graph. In this paper, we study the minimal embedding of the line graph associated to Γ(R), denoted by L(Γ(R)), into compact surfaces (orientable or non-orientable) and completely classify all finite commutative rings R such that the line graphs associated to their zero-divisor graphs have genera or crosscaps up to two.     

Eigenvalues of zero-divisor graphs of finite commutative rings

Journal of Algebraic Combinatorics - We investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$ of finite commutative rings R and study the interplay between these eigenvalues, the...     

每一个元素都是左零因子之环

本文引进左(右)零因子环的概念,它们是一类无单位元的环.我们称一个环为左(右)零因子环,如果对于任何 $a \in R$,都有$r_R (a) \neq 0~(l_R(a)\neq 0)$,而称一个环为强左(右)零因子环,如果$r_R(R)\neq 0~(l_R(R)\neq 0)$.Camillo和Nielson称一个环$R$为右有限零化环(简称RFA-环),如果$R$的每一个有限子集都有非零的右零化子.本文给出左零因子环的一些基本例子,探讨强左零因子环和RFA-环的扩张,并给出它们的等价刻画.     

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)?{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.     

The Zero-Divisor Graph Associated to a Semigroup

The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and quotient sets. We also prove necessary and sufficient conditions for determining when regular graphs and complete graphs with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that satisfy all known necessary conditions but are not zero-divisor graphs.     

Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.