Extending the Anderson-Livingston zero-divisor graph via unfaithful modules

Given a commutative ring R with unity and a left R -module M , the graph \Gamma(R, M) is defined on the set of nonzero zero-divisors of R modulo \operatorname{Ann}_R(M) . A vertex r lies in \Gamma(R, M) if r \in R - \operatorname{Ann}_R(M) : r r' = 0 \pmod{\operatorname{Ann}_R(M)} for some r' \in R - \operatorname{Ann}_R(M) . Two distinct vertices r and s are adjacent if r s M = 0 . This graph generalizes the Anderson?Livingston zero-divisor graph \Gamma(R) . We study its structure and establish formulas that link the invariants of \Gamma(R, M) and \Gamma(R) .