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INDECOMPOSABILITY GRAPHS OF RINGS

Part of: Conditions on elements Graph theory

Published online by Cambridge University Press:  01 February 2008

KARIN CVETKO-VAH  and
DAVID DOLŽAN
KARIN CVETKO-VAH
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia
DAVID DOLŽAN
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia (email: [email protected])
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Abstract

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We define a subgraph of the zero divisor graph of a ring, associated to the ring idempotents. We study its properties and prove that for large classes of rings the connectedness of the graph is equivalent to the indecomposability of the ring and in those cases we also calculate the graph’s diameter.

Keywords

ringgraphidempotentzero divisor

MSC classification

Secondary: 16U99: None of the above, but in this section 05C99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 1 , February 2008 , pp. 151 - 159
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Akbari, S. and Mohammadian, A., ‘Zero-divisor graphs of non-commutative rings’, J. Algebra 296 (2006), 462–479.CrossRefGoogle Scholar
[2]Beck, I., ‘Coloring of a commutative ring’, J. Algebra 116 (1988), 208–226.CrossRefGoogle Scholar
[3]Benkart, G. M. and Osborn, J. M., ‘Derivations and automorphisms of nonassociative matrix algebras’, Trans. Amer. Math. Soc. 263(2) (1981), 411–430.CrossRefGoogle Scholar
[4]Hirano, Y. and Sumiyama, T., ‘On orders of directly indecomposable finite rings’, Bull. Austral. Math. Soc. 46(3) (1992), 353–359.CrossRefGoogle Scholar
[5]Mainwaring, D. and Pearson, K. R., ‘Decomposability of finite rings’, J. Austral. Math. Soc. Ser. A 28 (1979), 136–138.CrossRefGoogle Scholar
[6]Redmond, S. P., ‘The zero-divisor graph of a non-commutative ring’, Internat. J. Commutative Rings 1(4) (2002), 203–211.Google Scholar