The i-extended zero-divisor graphs of idealizations.

Let R be a commutative ring with zero-divisors Z (R) and i be a positive integer. The i -extended zero-divisor graph of R , denoted by Γ ¯ i (R) , is the (simple) graph with vertex set Z (R) ∗ = Z (R) \ { 0 } , the set of nonzero zero-divisors of R , and two distinct nonzero zero-divisors x and y are adjacent whenever there exist two positive integers n , m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. The i -extended zero-divisor graph of R is well studied in 10. In this paper, we characterize the i -extended zero-divisor graphs of idealizations R ⋉ M (where M is an R -module). Namely, we study in detail the behavior of the filtration ( Γ ¯ i (R ⋉ M)) i ∈ ℕ ∗ as well as the relations between its terms. We also characterize the girth and the diameter of Γ ¯ i (R ⋉ M) and we give answers to several interesting and natural questions that arise in this context. [ABSTRACT FROM AUTHOR]