A non-parallel submodules graph of M, denoted by G∦(M), is an undirected simple graph
whose vertices are in one-to-one correspondence with all non-zero proper submodules of M …

We show that an $ R $-module $ M $ is noetherian (resp., artinian) if and only if its
annihilating submodule graph, $\mathbb {G}(M) $, is a non-empty graph and it has …

Let R be a ring with identity and M be a unitary left R-module. The complement of the
intersection graph of submodules of M, denoted by Γ (M), is defined to be a graph whose …

The intersection graph of nontrivial submodules of M, denoted by G (M), is an undirected
simple graph whose vertices are in one-to-one correspondence with all nontrivial …

Let $ R $ be a ring and $ M $ be an $ R $-module. Two modules $ A $ and $ B $ are called
orthogonal, written $ A\perp B $, if they do not have non-zero isomorphic submodules. We …

Let M be a unitary left R-module, where R is a (not necessarily commutative) ring with
identity. The small intersection graph of nontrivial submodules of M, denoted by Γ (M), is an …

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-
module M, denoted by G (M), is defined to be the undirected simple graph whose vertices …

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 …

Let R be a ring with identity and M be a unitary left R-module. The inclusion submodule
graph of a module M, denoted by In (M), is an undirected simple graph whose vertex set V …

‌In this paper we continue our study of perpendicular graph of modules‌, ‌that was introduced
in\cite {Hokkaido} ‌. ‌Let $ R $ be a ring and $ M $ be an $ R $-module‌. ‌Two modules $ A …