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
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.
[1] Z. Abdollah, P. Malakooti, Sh. Ghalandarzadeh, Sh. Shahriari, On torsion elements and their annihilators, J. Algebra. 610 (2022), 199-222.
[2] D. F. Anderson, T. Asir, A. Badawi, T. Tamizh Chelvam, Graphs from Rings, Springer, Cham, 2021.
[3] D. F. Anderson, M. C. Axtell, J. A. Stickles, Zero-divisor graphs in commutative rings, Commutative Algebra-Noetherian and Non-Noetherian Perspectives, Springer, New York, 2011.
[4] D. F. Anderson, Ph. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra. 217 (2) (1999), 434-447.
[5] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra. 42 (1) (2014), 108-121.
[6] A. Barnard, Multiplication modules, J. Algebra. 71 (1) (1981), 174-178.
[7] I. Beck, Coloring of commutative rings, J. Algebra. 116 (1) (1988), 208-226.
[8] F. DeMeyer, K, Schneider, Automorphisms and Zero Divisor Graphs of Commutative Rings, Commutative Rings, Nova Sci. Publ. NY, 2002.
[9] Z. Abd El-Bast, P. F. Smith, Multiplication modules, Comm. Algebra. 16 (4) (1988), 755-779.
[10] Sh. Ghalandarzadeh, P. Malakooti Rad, Torsion graph over multiplication modules, Extracta Math. 24 (3) (2009), 281-299.
[11] Sh. Ghalandarzadeh, P. Malakooti Rad, Torsion graph of modules, Extracta Math. 26 (1) (2011), 153-163.
[12] T. K. Lee, Y. Zhou, Reduced modules, rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math. 236 (2004), 365-377.
[13] Ch. P. Lu, Prime submodules of modules, Rikkyo Daigaku Sugaku Zasshi. 33 (1) (1984), 61-69.
[14] F. Mehdi, On multiplication modules, Math. Student. 42 (1974), 149-153.