The weakly zero-divisor graph of a commutative ring

Abstract

. Let R be a commutative ring with identity, and let Z ( R ) be the set of zero-divisors of R . The weakly zero-divisor graph of R is the undirected (simple) graph W Γ( R ) with vertex set Z ( R ) ∗ , and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann( x ) and s ∈ ann( y ) such that rs = 0. It follows that W Γ( R ) contains the zero-divisor graph Γ( R ) as a subgraph. In this paper, the connectedness, diameter, and girth of W Γ( R ) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of W Γ( R ) is studied.