The i-extended zero-divisor graphs of idealizations

Abstract

Let R be a commutative ring with zero-divisors [Formula: see text] and [Formula: see text] be a positive integer. The [Formula: see text]-extended zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is the (simple) graph with vertex set [Formula: see text], the set of nonzero zero-divisors of [Formula: see text], and two distinct nonzero zero-divisors [Formula: see text] and [Formula: see text] are adjacent whenever there exist two positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text] and [Formula: see text]. The [Formula: see text]-extended zero-divisor graph of [Formula: see text] is well studied in 10. In this paper, we characterize the [Formula: see text]-extended zero-divisor graphs of idealizations [Formula: see text] (where [Formula: see text] is an [Formula: see text]-module). Namely, we study in detail the behavior of the filtration [Formula: see text] as well as the relations between its terms. We also characterize the girth and the diameter of [Formula: see text] and we give answers to several interesting and natural questions that arise in this context.