What question did this study set out to answer?

The central aim is to explore the relationships between the properties of modules and their associated perpendicular graphs.

June 14, 2026Open Access

Further Studies of the Perpendicular Graphs of Modules

MSMaryam SHIRALIYasouj University SSSaeid SafaeeyanYasouj University

Key Points

The central aim is to explore the relationships between the properties of modules and their associated perpendicular graphs.
The study involves defining orthogonal modules and constructing perpendicular graphs associated with $R$-modules.
Key graph properties, such as clique number and chromatic number, are analyzed in relation to the modules' properties.
The paper demonstrates results using different types of modules, focusing particularly on semi-simple modules.
If the clique number of the graph $orall(ot)(M)$ is finite and $M$ has a simple submodule, then the chromatic number is also finite.
For semi-simple modules, it is established that the clique number equals the chromatic number, i.e., $orall(ot)(R M)=orall(ot)(R M)$ for such modules.

Abstract

‎In this paper we continue our study of perpendicular graph of modules‎, ‎that was introduced in Hokkaido‎. ‎Let R be a ring and M be an R-module‎. ‎Two modules A and‎ ‎B are called orthogonal‎, ‎written A B‎, ‎if they do not have‎ ‎non-zero isomorphic submodules‎. ‎We associate a graph _ (M) to M‎ ‎with vertices‎ ‎M_=\ (0) A M\;|\; (0) B M \; such that\; A B\‎, ‎and for distinct A, B‎ ‎M_‎, ‎the vertices A and B are adjacent if and only if‎ ‎A B‎. ‎The main object of this article is to study the‎ ‎interplay of module-theoretic properties of M with‎ ‎graph-theoretic properties of _ (M) ‎. ‎We study the clique number and chromatic number of _ (M) ‎. ‎We prove that if (_ (M) ) < and M has a simple submodule‎, ‎then (_ (M) ) < ‎. ‎Among other results‎, ‎it is shown that for a semi-simple module M‎, ‎ (_ (R M) ) = (_ (R M) ) ‎.

KI fragen

Bookmark

View Full Paper