On the compressed essential graph of a commutative ring

Abstract

Let R be a commutative ring. In this paper, we introduce and study the compressed essential graph of R, EG(E)(R). The compressed essential graph of R is a graph whose vertices are equivalence classes of non-zero zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(xy) is an essential ideal of R. It is shown if R is reduced, then EG(E)(R) = Gamma(E)(R), where Gamma(E)(R) denotes the compressed zero-divisor graph of R. Furthermore, for a non-reduced Noetherian ring R with 3 < vertical bar EG(E)(R)vertical bar < infinity, it is shown that EG(E)(R) = Gamma(E)(R) if and only if (i) Nil(R) = ann(Z(R)). (ii) Every non-zero element of Nil(R) is irreducible in Z(R).