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International Journal of Mathematics and Mathematical Sciences
Volume 17, Issue 3, Pages 463-468
http://dx.doi.org/10.1155/S0161171294000670

Finite completely primary rings in which the product of any two zero divisors of a ring is in its coefficient subring

Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 6 June 1990; Revised 30 August 1993

Copyright © 1994 Hindawi Publishing Corporation. 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

According to general terminology, a ring R is completely primary if its set of zero divisors J forms an ideal. Let R be a finite completely primary ring. It is easy to establish that J is the unique maximal ideal of R and R has a coefficient subring S (i.e. R/J isomorphic to S/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is in S and determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary rings with either J2=0 or their coefficient subring is Z2n with n=2 or 3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.