The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

Classifying annihilating-ideal graphs of commutative artinian rings

In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [10 Behboodi, M., Rakeei, Z. (2011). The annihilating-ideal graph of commutative rings I. J. Algebra Appl. 10(4):727–739.[Crossref], [Web of Science ®] , [Google Scholar]]. Our main goal is to determine which algebraic properties of a ring are reflected in its annihilating-ideal graph. We prove that, for artinian rings, the annihilating-ideal graph can be used to determine whether the ring in question is a PIR or, more generally, if it is a dual ring. Moreover, with one trivial exception, the annihilating-ideal graph can distinguish between PIRs with different ideal lattices. In addition, we explore new techniques for classifying small annihilating-ideal graphs. Consequently, we completely determine the graphs with six or fewer vertices which can be realized as the annihilating-ideal graph of a commutative ring.     

有单位元交换环上二阶矩阵的因子分解

对有单位元交换环上矩阵分解问题进行了讨论,给出了有单位元交换环上二阶矩阵可以因式分解的充分必要条件,即单位元交换环上二阶矩阵可以因式分解当且仅当这个矩阵的行列式可以因子分解.     

Similarity invariants for matrices over a commutative Artinian chain ring

Suppose that R is a commutative Artinian chain ring, A is an m × m matrix over R, and S is a discrete valuation ring such that R is a homomorphic image of S. We consider m ideals in the polynomial ring over S that are similarity invariants for matrices over R, i.e., these ideals coincide for similar matrices. It is shown that the new invariants are stronger than the Fitting invariants, and that new invariants solve the similarity problem for 2 × 2 matrices over R.     

The nonexistence of sensitive commutative group actions on graphs

In this paper,we observe a special kind of continuous functions on graphs.By estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on graphs.Furthermore,we consider a 1-dimensional continuum composed of a spiral curve and a circle and show that there exist sensitive homeomorphisms on it,which answers negatively a question proposed by Kato in 1993.     

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.     

交换环上特殊线性李代数的极大子代数

文章利用有单位元且2,3是单位的交换环的极大理想刻画了其上特殊线性李代数包含典范环面的极大子代数.确定了特殊线性李代数极大子代数的个数,并证明了每个极大子代数均可通过置换矩阵共轭于标准的极大子代数.     

On idempotents of a class of commutative rings

Abstract

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results.     

The range of a ring homomorphism from a commutative -algebra

We prove that if a commutative semi-simple Banach algebra is the range of a ring homomorphism from a commutative -algebra, then is -equivalent, i.e. there are a commutative -algebra and a bicontinuous algebra isomorphism between and . In particular, it is shown that the group algebras , and the disc algebra are not ring homomorphic images of -algebras.

     

Noether环理想的性质

Noether交换环是一类非常重要的环,本文主要对Noether交换环进行了研究和讨论,得到了Noether交换环、Noether整环的若干性质;并推广了文[1]中的部分结果.     

Abelian正则环的零因子图

We introduce the zero-divisor graph for an abelian regular ring and show that if R, S are abelian regular, then (K0(R),[R])≌(K0(S),[S])if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular,moreover,the zero-divisor graph of such a ring is studied.     

可换环上一般线性李代数在几类典型李代数中的扩代数

研究典型李代数的子代数结构,利用矩阵方法决定了含幺可换环上n级一般线性李代数分别在2n级辛代数,2n级正交代数及2n 1级正交代数中的扩代数.