<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

Online veröffentlicht: 22. Dez. 2015

Seitenbereich: 127 - 136

Eingereicht: 12. Juni 2013

Akzeptiert: 24. Sept. 2013

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

Schlüsselwörter
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

Online veröffentlicht: 22. Dez. 2015

Seitenbereich: 127 - 136

Eingereicht: 12. Juni 2013

Akzeptiert: 24. Sept. 2013

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

SchlüsselwörterDiameter, 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.

Schlüsselwörter
Diameter, Girth, Multiplication modules, Torsion graphs