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International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 1, Pages 121-126
http://dx.doi.org/10.1155/S0161171279000120

Rings with a finite set of nonnilpotents

1Department of Mathematics, N.C. State University, Raleigh 27607, N.C., USA
2Department of Mathematics, University of California, Santa Barbara 93106, California, USA

Received 16 October 1978

Copyright © 1979 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

Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn-ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.