Undirected Zero-Divisor Graphs and Unique Product Monoid Rings

• 来源期刊: 代数集刊（英文版）
• 关键词: zero-divisor graph diameter semi-commutative ring unique product monoid monoid ring
• 年，卷（期）: 2019,（4）
• 页码范围: 665-676
• 页数: 12页
• 语种: 英文
• DOI: 10.1142/S1005386719000488

摘要

Let R be an associative ring with identity and Z* (R) be its set of non-zero zero-divisors.The undirected zero-divisor graph of R,denoted by Γ(R),is the graph whose vertices are the non-zero zero-divisors of R,and where two distinct vertices r and s are adjacent if and only if rs =0 or sr =0.The distance between vertices a and b is the length of the shortest path connecting them,and the diameter of the graph,diam(F(R)),is the superimum of these distances.In this paper,first we prove some results about Γ(R) of a semi-commutative ring R.Then,for a reversible ring R and a unique product monoid M,we prove 0 ≤ diam(Γ(R)) ≤ diam(Γ(R[M])) ≤ 3.We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])),strictly in terms of the properties of a ring R,where R is a reversible ring and M is a unique product monoid.Moreover,an example showing the necessity of our assumptions is provided.