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Journal of Mathematics
Volume 2013 (2013), Article ID 467905, 8 pages
http://dx.doi.org/10.1155/2013/467905
Research Article

Classification of Finite Rings of Order by Generators and Relations

1Department of Mathematics, Islamic Azad University, Khorramabad Branch, Khorramabad 68178-16645, Iran
2Department of Mathematics, Lorestan University, P.O. Box 465, Khorramabad 68137-17133, Iran
3Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord 88186-34141, Iran

Received 26 September 2012; Accepted 10 December 2012

Academic Editor: Liwei Zhang

Copyright © 2013 P. Karimi Beiranvand et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For any finite abelian group , we define a binary operation or “multiplication” on and give necessary and sufficient conditions on this multiplication for to extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there are rings of order with characteristic , where is a prime number. Also, all finite rings of order are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.