Star graphs for torsion elements in multiplication modules

Subject Areas : Commutative algebra

Z. ‎Abdollah ¹ , P. ‎Malakooti Rad ² , Sh. Ghalandarzadeh ³ , Sh. ‎Shahriari ⁴

1 - Department of Mathematics‎, ‎Qazvin Branch‎, ‎Islamic Azad University‎, ‎Qazvin‎, ‎Iran
2 - Department of Mathematics‎, ‎Qazvin Branch‎, ‎Islamic Azad University‎, ‎Qazvin‎, ‎Iran
3 - Faculty of Mathematics‎, ‎K‎. ‎N‎. ‎Toosi University of Technology‎, ‎Tehran‎, ‎Iran
4 - Department of Mathematics & Statistics‎, ‎Pomona College‎, ‎Claremont‎, ‎CA 91711‎, ‎USA

Received: 2024-04-29 Accepted : 2024-09-03 Published : 2024-09-04

Keywords: Annihilator graphs‎, ‎zero-divisor graphs‎, ‎star graphs‎, ‎torsion elements‎, ‎annihilators‎, ‎modules‎, ‎multiplication modules‎, ‎reduced modules,

Abstract :

‎Let $R$ be a commutative ring with identity‎, ‎$M$ a multiplication $R$-module‎, ‎and $T(M)^\star$ the set of non-zero torsion elements of $M$‎. ‎We consider two graphs‎, ‎the torsion graph and the annihilator graph of $M$ that have $T(M)^\star$ as their set of vertices‎, ‎and investigate the cases when these graphs are stars‎. ‎The graph theoretic properties are reflected in the ring theoretic properties and vice versa‎. ‎If a ring is considered as a module on itself‎, ‎then the module is a multiplication module‎. ‎Hence‎, ‎our results directly generalize results about rings‎.

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