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September 18, 2025Open Access

Minimal submodules graph of modules over commutative rings

DMDwi Mifta MahananiUniversity of Brawijaya DIDewi IsmiartiUniversitas Islam Negeri Maulana Malik Ibrahim SASyaiful AnamUniversity of Brawijaya

Key Points

The study introduces the minimal submodules graph, enhancing understanding of module structures and relationships.
Adjacent vertices in the graph indicate minimal intersections, highlighting intrinsic qualities of $R$-modules.
Properties of the graph reflect those of the underlying module, suggesting a deep interplay between algebraic structures.
Comparative analysis of $ ext{min}$ invariant numbers reveals distinctions between the graphs of different modules.

Abstract

Let R be a commutative ring and M be an R-module. In this paper, we define minimal submodules graph of M, denoted by ₌₈₍ (M), in which the vertex set is the set of nonzero proper submodules of M. Two distinct vertices A and B are adjacent provided that A B is a minimal submodule of M. In this study, we associate some properties of the graph from the properties of module and vice versa. Moreover, if we have an R-module homomorphism from M to M', we compare some invariant numbers and properties of ₌₈₍ (M) and ₌₈₍ (M').

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