Star graphs for torsion elements in multiplication modules
محورهای موضوعی : Commutative algebra
Z. Abdollah
1
(Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran)
P. Malakooti Rad
2
(Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran)
Sh. Ghalandarzadeh
3
(Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran)
Sh. Shahriari
4
(Department of Mathematics & Statistics, Pomona College, Claremont, CA 91711, USA)
کلید واژه: Annihilator graphs, zero-divisor graphs, star graphs, torsion elements, annihilators, modules, multiplication modules, reduced modules,
چکیده مقاله :
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.
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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