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 ∈ M is a vertex if there exists 0 M ≠ P ∈ M such that N P = 0 M and two vertices N 1 , N 2 are adjacent if N 1 N 2 = 0 M (where N 1 N 2 = N 1 : I M N 2 : I M 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 R G M is isomorphic to a subgraph of Zariski topology graph Ģ X M with respect to X .

Prime Spectrum Graph of C-Lattices

“…According to I. Beck, the zero-divisor graph of a commutative ring is a graph where the vertices represent the elements of the ring, and two vertices are connected by an edge if their product is zero (see [5]). Further, by employing these graph-theoretic approach, many researchers visually represented and analyzed various aspects of commutative rings, providing additional insights into their structure, properties, and relationships (see [1]- [4], [9]- [10], [12]).…”