The Compressed Annihilator Graph of a Commutative Ring

Abstract

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AGE(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ⊂ ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.