The n-zero-divisor graph of a commutative semigroup

Abstract

Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph with vertices and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph with vertices and distinct vertices x and y are adjacent if and only if xy = 0. Thus each is an induced subgraph of We pay particular attention to and the case when S is a commutative ring with We also consider several other types of “n-zero-divisor” graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent.