## SUBMODULE INCLUSION GRAPH OF A MODULE

## J. Goswami

DOI: https://doi.org/10.37418/amsj.9.11.98

Let $R$ be a commutative ring with unity and $M$ be an $R$-module. The Submodule inclusion graph of $M$, denoted by $I_{S}(M)$, is a (undirected) graph with vertices as all non-trivial submodules of $M$ and two distinct vertices $N$ and $L$ are adjacent if and only if $N \subset L$ or $L \subset N$. In this paper, it has been proved that $I_{S}(M)$ is not connected if and only if $M$ is a direct sum of two simple $R$-modules. Moreover, it has been shown that $I_{S}(M)$ is a complete graph if and only if $M$ is a uniserial module. The diameter, girth, clique number, and chromatic number of $I_{S}(M)$ have been studied. Finally, it has been shown that Beck’s Conjecture holds in $I_{S}(M)$ under certain condition.

**Keywords: **Commutative Ring, Inclusion Submodule Graph, Module, Submodule.