Zero Intersection Graph of Annihilator Ideals of Modules

Year 2025, Early Access, 1 - 13

Seçil Çeken , Osama A. Naji , Ünsal Tekir , Suat Koç

https://doi.org/10.15672/hujms.1485903

Abstract

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let R be a commutative ring having a nonzero identity and M be a nonzero unital R-module. The zero intersection graph of annihilator ideals of R-module M, denoted by CR(M), is a simple (undirected) graph whose vertex set M⋆ = M − {0}, and two distinct vertices m and m′ are adjacent if annR(m)∩annR(m′) = (0). We investigate the conditions under which CR(M) is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property T in terms of their graphical properties.

Keywords

Property A , Property T , torsion free modules , quasi regular ring , weak quasi regular module

References

• [1] S. Akbaria, H.A. Tavallaeeb and S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl. 10, 1-8, 2011