<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>R</italic> be a commutative ring with identity. Let <italic>M</italic> be an <italic>R</italic>-module and <italic>T</italic> (<italic>M</italic>)* be the set of nonzero torsion elements. The set <italic>T</italic>(<italic>M</italic>)* makes up the vertices of the corresponding torsion graph, Γ<italic>R</italic>(<italic>M</italic>), with two distinct vertices <italic>x, y</italic> ∈ <italic>T</italic>(<italic>M</italic>)* forming an edge if <italic>Ann</italic>(<italic>x</italic>) ∩ <italic>Ann</italic>(<italic>y</italic>) ≠ 0. In this paper we study the case where the graph Γ<italic>R</italic>(<italic>M</italic>) is connected with <italic>diam</italic>(Γ<italic>R</italic>(<italic>M</italic>)) ≤ 3 and we investigate the relationship between the diameters of Γ<italic>R</italic>(<italic>M</italic>) and Γ<italic>R</italic>(<italic>R</italic>). Also we study girth of Γ<italic>R</italic>(<italic>M</italic>), it is shown that if Γ<italic>R</italic>(<italic>M</italic>) contains a cycle, then <italic>gr</italic>(Γ<italic>R</italic>(<italic>M</italic>)) = 3.</abstract>

Let R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the corresponding torsion graph, ΓR(M), with two distinct vertices x, y ∈ T(M)* forming an edge if Ann(x) ∩ Ann(y) ≠ 0. In this paper we study the case where the graph ΓR(M) is connected with diam(ΓR(M)) ≤ 3 and we investigate the relationship between the diameters of ΓR(M) and ΓR(R). Also we study girth of ΓR(M), it is shown that if ΓR(M) contains a cycle, then gr(ΓR(M)) = 3.

Diameter and girth of Torsion Graph

Department of Mathematics, Islamic Azad University, Qazvin Branch, Qazvin, Iran.

School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology, Tehran, Iran.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Diameter and girth of Torsion Graph"}

Let R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the...

Diameter and girth of Torsion Graph

Published Online: Dec 22, 2015

Page range: 127 - 136

Received: Jun 12, 2013

Accepted: Sep 24, 2013

DOI: https://doi.org/10.2478/auom-2014-0054

Keywords
Diameter, Girth, Multiplication modules, Torsion graphs

© 2014 P. Malakooti Rad et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Diameter and girth of Torsion Graph

Published Online: Dec 22, 2015

Page range: 127 - 136

Received: Jun 12, 2013

Accepted: Sep 24, 2013

DOI: https://doi.org/10.2478/auom-2014-0054

KeywordsDiameter, Girth, Multiplication modules, Torsion graphs

© 2014 P. Malakooti Rad et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Diameter, Girth, Multiplication modules, Torsion graphs