International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 3, Pages 463-468
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 Yousif Alkhamees. 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.
According to general terminology, a ring is completely primary if its set of zero divisors forms an ideal. Let be a finite completely primary ring. It is easy to establish that is the unique maximal ideal of and has a coefficient subring (i.e. isomorphic to ) 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 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 or their coefficient subring is with or . A special case of these rings is the class of finite rings, studied in , in which the product of any two zero divisors is zero.