On torsion elements and their annihilators

Let R be a commutative ring with identity, and let M be an R-module. In this paper, we focus on the ideals of R that are annihilators of torsion elements of M. In analogy with definitions and results on zero-divisor and annihilator graphs of rings, we define the annihilator graph of a module. We investigate the structure, the diameter, and the girth of this graph and the closely related torsion graph of a module introduced by Ghalandarzadeh and Malakooti Rad. Given the right definitions, the properties of the modules are reflected in the graph theoretic properties of the graphs. We thus modify and extend results on zero divisors of rings to the much more general setting of modules and their torsion elements. In addition, we significantly strengthen the known results on the torsion graphs of modules. Some of the results will be refined further for the cases when M is a multiplication module or a reduced module.(c) 2022 Elsevier Inc. All rights reserved.