Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for Γ(RM). We Will show that Γ(RM) is connected withdiam Γ(RM)≤ 3 and if Γ(RM) contains a cycle, then Γ(RM)≤4. We will also show tha...

We prove a generalized version of Powers' averaging property that characterizes simplicity reduced crossed products $C(X) \rtimes_\lambda G$, where $G$ is countable discrete group, and $X$ compact Hausdorff space which acts on minimally by homeomorphisms. As consequence, we generalize results Hartman Kalantar unique stationarity to the state G$ Kawabe's amenable subgroups $\operatorname{Sub}_a(...