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THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS

  • Saraei, Fatemeh Esmaeili Khalil (Faculty of Fouman College of Engineering University of Tehran)
  • Received : 2013.07.27
  • Published : 2014.07.01

Abstract

Let M be a module over a commutative ring R, and let T(M) be its set of torsion elements. The total torsion element graph of M over R is the graph $T({\Gamma}(M))$ with vertices all elements of M, and two distinct vertices m and n are adjacent if and only if $m+n{\in}T(M)$. In this paper, we study the basic properties and possible structures of two (induced) subgraphs $Tor_0({\Gamma}(M))$ and $T_0({\Gamma}(M))$ of $T({\Gamma}(M))$, with vertices $T(M){\backslash}\{0\}$ and $M{\backslash}\{0\}$, respectively. The main purpose of this paper is to extend the definitions and some results given in [6] to a more general total torsion element graph case.

Keywords

total graph;torsion prime submodule;T-reduced

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