Torsion graph of modules
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2009
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Maltepe Üniversitesi
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Abstract
Let R be a commutative ring and M be an R-module. the concept of zero-divisor graph of a commutative ring was introduced by I. Beck in 1988. He let all elements of the ring be vertices of the graph and was interested mainly in colorings. In this talk, we give a generalization of the concept of zero-divisor graph in a commutative ring with identity to torsiongraph in a module. We associate to M a graph denoted by ?(M) called torsion graph of M whose vertices are non-zero torsion elements of M and two different elements x, y ? T(M) ? {0} are adjacent if and only if [x : M][y : M]M = 0. The residual of Rx by M, denoted by [x : M], is a set of elements r ? R such that rM ? Rx for x ? M. The annihilator of an R-module M denoted by AnnR(M) is [0 : M]. Let T(M) be a set of element of M such that Ann(m) 6= 0. It is clear that if R be an integral domain T(M) is a submodue of M and is called torsion submodule of M. We investigate the interplay between module-theoretic properties of M and the graph-theoretic properties of ?(M). An R-module M is a multiplication module if for every R-submodule K of M there is an ideal I of R such that K = IM. Among the other result, we prove that ?(M) is finite if and only if either M is finite or M is a torsion free R-module and ?(M) is connected and diam(?(M)) ? 3 for faithful R-module M, and that if M be a multiplication R-module. then there is a vertex of ?(M) which is adjacent to every other vertex if and only if either M = M1 ?M2 is a faithful R-module, where M1, M2 are two submodules of M such that M1 has only two elements, M2 is finitely generated with T(M) = {(x, 0), (0, m2)|x ? M1, m2 ? M2}, or T(M) = IM, where I is an annihilator ideal of R. Also if M be a multiplication R-module, then ?(M) and ?(S ?1M) are isomorphic as graph where S = R ? Z(M).
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International Conference of Mathematical Sciences
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Citation
Rad, P. M., Ghalandarzadeh, S. ve Shirinkam S. (2009). Torsion graph of modules. Maltepe Üniversitesi. s. 322.
