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Rings having zero-divisor graphs of small diameter or large girth

Published online by Cambridge University Press:  17 April 2009

S. B. Mulay
S. B. Mulay
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, United States of America, e-mail: mulay@math.utk.edu
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Let R be a commutative ring possessing (non-zero) zero-divisors. There is a natural graph associated to the set of zero-divisors of R. In this article we present a characterisation of two types of R. Those for which the associated zero-divisor graph has diameter different from 3 and those R for which the associated zero-divisor graph has girth other than 3. Thus, in a sense, for a generic non-domain R the associated zero-divisor graph has diameter 3 as well as girth 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 481 - 490
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Anderson, D.F. and Livingston, P.S., ‘The zero-divisor graph of a commutative ring,’ J. Algebra 217 (1999), 434–447.Google Scholar
[2]Anderson, D.F., Levy, R. and Shapiro, J., ‘Zero-divisor graphs, von Neumann regular rings and Boolean algebras,’ J. Pure Appl. Algebra 180 (2003), 221–241.Google Scholar
[3]Beck, I., ‘Coloring of commutative rings,’ J. Algebra 116 (1988), 208–226.CrossRefGoogle Scholar
[4]Mulay, S.B., ‘Cycles and symmetries of zero-divisors,’ Comm, Algebra 30 (2002), 3533–3558.CrossRefGoogle Scholar
[5]Nagata, M., Local rings (Krieger Publishing Company, Huntington, N.Y., 1975).Google Scholar