On Finite Factorization Rings

Let R be a commutative ring with identity. R is a finite factorization ring (FFR) if every nonzero nonunit of R has only a finite number of factorizations up to order and associates. In this article, we give a characterization of R for R[X] and R[[X]] to be an FFR.     

Lattice-Ordered Matrix Rings Over the Integers

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.     

Polynomial Rings Over Symmetric Rings Need Not be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

The Total Graphs of Finite Rings

In this paper we extend the study of total graphs τ(R) to noncommutative finite rings R. We prove that τ(R) is connected if and only if R is not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite rings R.     

交换环上多项式结式的性质

本文主要讨论交换环上多项式结式的一些性质.首先,我们证明了交换环上一种乘积的结式等于结式的乘积的性质,然后,我们证明了交换环上一种和的结式具有的性质,并且给出了交换环上结式为零的一个充分条件.     

平坦的多项式剩余类环

本文证明了如果多项式的剩余类环 A=R[T]/fR[T]作为 R-模是平坦模,且R是约化环,则f是正规多项式.特别地,若R还是连通的,则f的首项系数是单位.也证明了弱整体有限的凝聚环是约化环,以及弱整体为有限的凝聚连通环是整环.     

Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

ABSTRACT

We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power series rings.     

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.     

Irreducible Divisor Graphs and Factorization Properties of Domains

This article examines the connections between the factorization properties of a domain, e.g., unique factorization domain (UFD), finite factorization domain (FFD), and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x): x ∈ D*  U(D)} when D is an FFD, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x): x ∈ D*  U(D)} when D is not an FFD. We also introduce an alternate irreducible divisor graph called the compressed irreducible divisor graph and study some of its properties.     

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ? ?₂ × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.     

A Note on Invariants Over Finite Local Rings

In this note, we prove a conjecture of Steinberg [10 Steinberg , R. (1987). On Dickson's theorem on invariants. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34:699–707. [Google Scholar]] about modular invariants of some linear groups over finite commutative local rings.     

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).     

Galois环上的本原多项式的一个判别准则

本文给出Ｇａｌｏｉｓ环Ｒ上的基本不可约多项式ｆ（ｘ）的根的具体表达式和其阶的联系；由此，对本原多项式和次本原多项式分别推导出代数判别式，其主要部分分别由ｆ（ｘ）ｍｏｄｐ和ｆ（ｘ）ｍｏｄｐ２的系数所确定．     

Strong Cleanness of Matrix Rings Over Commutative Rings

Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring 𝕄 _n (R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is a valuation ring. It is also shown that each R-algebra which is locally a direct limit of module-finite algebras, is strongly clean if R is a π-regular commutative ring.     

Cut Vertices in Zero-Divisor Graphs of Finite Commutative Rings

A cut vertex of a connected graph is a vertex whose removal would result in a graph having two or more connected components. We examine the presence of cut vertices in zero-divisor graphs of finite commutative rings and provide a partial classification of the rings in which they appear.     

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

On Least Common Multiples of Polynomials in Z/n Z[x]

Let 𝒫(n, D) be the set of all monic polynomials in ?/n?[x] of degree D. A least common multiple for 𝒫(n, D) is a monic polynomial L ∈ ?/n?[x] of minimal degree such that f divides L for all f ∈ 𝒫(n, D). A least common multiple for 𝒫(n, D) always exists, but need not be unique; however, its degree is always unique. In this article, we establish some bounds for the degree of a least common multiple for 𝒫(n, D), present constructions for common multiples in ?/n?[x], and describe a connection to rings of integer-valued polynomials over matrix rings.     

The Structure of Finite Local Principal Ideal Rings

A ring R is called a principal ideal ring (PIR), if each ideal of R is a principal ideal. A local ring (R, 𝔪) is an artinian PIR if and only if its maximal ideal 𝔪 is principal and has finite nilpotency index. In this article, we determine the structure of a finite local PIR.