THE CONNECTED SUBGRAPH OF THE TORSION GRAPH OF A MODULE

Abstract

In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set <TEX>$T(M)^*$</TEX> makes up the vertices of the corresponding torsion graph, <TEX>${\Gamma}(M)$</TEX>, with any two distinct vertices forming an edge if <TEX>$[x:M][y:M]M=0$</TEX>. We prove that, if <TEX>${\Gamma}(M)$</TEX> contains a cycle, then <TEX>$gr({\Gamma}(M)){\leq}4$</TEX> and <TEX>${\Gamma}(M)$</TEX> has a connected induced subgraph <TEX>${\overline{\Gamma}}(M)$</TEX> with vertex set <TEX>$\{m{\in}T(M)^*{\mid}Ann(m)M{\neq}0\}$</TEX> and diam<TEX>$({\overline{\Gamma}}(M)){\leq}3$</TEX>. Moreover, if M is a multiplication R-module, then <TEX>${\overline{\Gamma}}(M)$</TEX> is a maximal connected subgraph of <TEX>${\Gamma}(M)$</TEX>. Also <TEX>${\overline{\Gamma}}(M)$</TEX> and <TEX>${\overline{\Gamma}}(S^{-1}M)$</TEX> are isomorphic graphs, where <TEX>$S=R{\backslash}Z(M)$</TEX>. Furthermore, we show that, if <TEX>${\overline{\Gamma}}(M)$</TEX> is uniquely complemented, then <TEX>$S^{-1}M$</TEX> is a von Neumann regular module or <TEX>${\overline{\Gamma}}(M)$</TEX> is a star graph.