Some results on the intersection graph of submodules of a module

Rok, strany: 2017, 297 - 304

Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The intersection graph of submodules of $M$, denoted by $G(M)$, is defined to be a graph whose vertices are in one to one correspondence with all non-trivial submodules of $M$ and two distinct vertices are adjacent if and only if the corresponding submodules of $M$ have non-zero intersection. In this paper, we consider the intersection graph of submodules of a module. We determine the structure of modules whose clique numbers are finite. We show that if $1<ω(G(M))<∞$, then $M$ is a direct sum of a finite module and a cyclic module, where $ω(G(M))$ denotes the clique number of $G(M)$. We prove that if $ω(G(M))$ is not finite, then $M$ contains an infinite clique. Among other results, it is shown that a Noetherian $R$-module whose intersection of all non-trivial submodules is non-zero, is Artinian.

ISO 690:

Akbari, S., Tavallaee, H., Ghezelahmad, S. 2017. Some results on the intersection graph of submodules of a module. In Mathematica Slovaca, vol. 67, no.2, pp. 297-304. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0267

APA:

Akbari, S., Tavallaee, H., Ghezelahmad, S. (2017). Some results on the intersection graph of submodules of a module. Mathematica Slovaca, 67(2), 297-304. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0267