The connected subgraph of the torsion graph of a module (Q2914424)

The concept of a zero-divisor graph of a commutative ring was introduced by \textit{I. Beck} in [J. Algebra 116, No. 1, 208--226 (1988; Zbl 0654.13001)], where he was mainly interested in colorings, which establishes a connection between graph theory and commutative ring theory. Since then, the concept of the zero-divisor graphs has been studied extensively by many authors. This concept has been extended to non-commutative rings by \textit{S. P. Redmond} [Trends in commutative rings research. Reprint. Hauppauge, NY: Nova Science Publishers. 203-211 (2004); reprint of Int. J. Commut. Rings 1, No. 4, 203--211 (2002; Zbl 1195.16038)], to semigroups by \textit{F. R. DeMeyer, T. McKenzie} and \textit{K. Schneider} [Semigroup Forum 65, No. 2, 206--214 (2002; Zbl 1011.20056)], and to near-rings by \textit{A. G. Cannon, K. M. Neuerburg} and \textit{S. P. Redmond} [Nearrings and nearfields. Proceedings of the conference on nearrings and nearfields, Hamburg, Germany, July 27--August 3, 2003. Dordrecht: Springer. 189--200 (2005; Zbl 1084.16038)]. This paper investigates the concept of the torsion-graph of a module, which has been defined by the first two authors in [Extr. Math. 24, No. 3, 281--299 (2009; Zbl 1203.13007)].NEWLINENEWLINEThe torsion graph \(\Gamma(M)\) of \(R\)- module \(M\) is a simple graph, whose vertices set is the set \(T(M)^*\) of the non-zero torsion elements of \(M\), and two distinct elements \(x; y\) are adjacent if and only if \([x : M][y : M]M = 0\). The paper prove that, if \(\Gamma(M)\) contains a cycle, then \(gr(\Gamma(M))\leq 4\) and \(\Gamma(M)\) has a connected induced subgraph \(\bar{\Gamma}(M)\) with vertex set \(\{m\in T(M)^*~|~ {\text{ Ann}}(m)M\neq 0\}\) and diam\((\bar{\Gamma}(M) )\leq 3\). Moreover, if \(M\) is a multiplication \(R\)-module, then \(\bar{\Gamma}(M)\) is a maximal connected subgraph of \(\Gamma(M)\). Also \(\bar{\Gamma}(M)\) and \(\bar{\Gamma}(S^{-1}M)\) are isomorphic graphs, where \(S=R\setminus Z(M)\). They also show that, if \(\bar{\Gamma}(M)\) is uniquely complemented, then \(S^{-1}M\) is a von Neumann regular module or \(\bar{\Gamma}(M)\) is a star graph.NEWLINENEWLINEThis study helps to illuminate the structure of the torsion submodule of \(M\).