On torsion elements and their annihilators (Q2675062)

Let \(R\) be a commutative ring with non-zero identity and let \(M\) be a unitary \(R\)-module. An element \(x\in M\) is called a torsion element if \(\mathrm{Ann}_R(x) = \{0_R\}\). The set of torsion elements of \(M\) is denoted by \(\mathrm{T}(M)\), and \(\mathrm{T}^*(M) = \mathrm{T}(M)- \{0_M\}\). The annihilator graph for an \(R\)-module \(M\) is denoted by \(\mathrm{AG}(M)\), the set \(\mathrm{T}^*(M)\) -- consisting of non-zero torsion elements of \(M\) -- will be the set of vertices, and \(x, y\in\mathrm{T}^*(M))\) will be adjacent if and only if \(\mathrm{Ann}_R([x : M]y) = \mathrm{Ann}_R(x)\cup \mathrm{Ann}_R(y)\), or \(\mathrm{Ann}_R([y : M]x) = \mathrm{Ann}_R(x) \cup \mathrm{Ann}_R(y)\). If \(M = R\), then, for the \(R\)-module \(M\), we have \(\mathrm{T}(M) = \mathrm{Z}(R)\), where \(\mathrm{Z}(R)\) is the set of zero divisors of \(R\). In the paper under review, the authors extend results on zero divisors of rings to the general setting of modules and their torsion elements. In addition, they 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.