THE CONNECTED SUBGRAPH OF THE TORSION GRAPH OF A MODULE

Title & Authors
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 $\small{T(M)^*}$ makes up the vertices of the corresponding torsion graph, $\small{{\Gamma}(M)}$, with any two distinct vertices forming an edge if $\small{[x:M][y:M]M=0}$. We prove that, if $\small{{\Gamma}(M)}$ contains a cycle, then $\small{gr({\Gamma}(M)){\leq}4}$ and $\small{{\Gamma}(M)}$ has a connected induced subgraph $\small{{\overline{\Gamma}}(M)}$ with vertex set $\small{\{m{\in}T(M)^*{\mid}Ann(m)M{\neq}0\}}$ and diam$\small{({\overline{\Gamma}}(M)){\leq}3}$. Moreover, if M is a multiplication R-module, then $\small{{\overline{\Gamma}}(M)}$ is a maximal connected subgraph of $\small{{\Gamma}(M)}$. Also $\small{{\overline{\Gamma}}(M)}$ and $\small{{\overline{\Gamma}}(S^{-1}M)}$ are isomorphic graphs, where $\small{S=R{\backslash}Z(M)}$. Furthermore, we show that, if $\small{{\overline{\Gamma}}(M)}$ is uniquely complemented, then $\small{S^{-1}M}$ is a von Neumann regular module or $\small{{\overline{\Gamma}}(M)}$ is a star graph.
Keywords
torsion graph;multiplication modules;von Neumann regular modules;
Language
English
Cited by
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