Residual Division Graph of Lattice Modules

Abstract

Let $L$ be a multiplicative lattice and $M$ be a lattice module over $L$ . In this paper, we assign a graph to $M$ called residual division graph RG(M) in which the element $N\in M$ is a vertex if there exists $0_M\ne P\in M$ such that $NP=0_M$ and two vertices $N_1,N_2$ are adjacent if $N_1{N}_2=0_M$ (where $N_1{N}_2=\left(N_1:I_M\right)\left(N_2:I_M\right)I_M$ ). It is proved that such a graph with the greatest element $I_M$ which does not belong to the vertex set is nonempty if and only if $M$ is a prime lattice module. Also, we provide conditions such that $RG\left(M\right)$ is isomorphic to a subgraph of Zariski topology graph ${Ģ}_X\left(M\right)$ with respect to $X$ .