Vol. 87, No. 2, 1980

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Generalized inverses in regular rings

Thomas R. Savage

Vol. 87 (1980), No. 2, 455–466

Motivated by I. Kaplansky’s theorem on one-sided inverses in rings, we consider, for a given nonzero element a in a regular ring, the number of solutions x to (i) a = axa, (ii) a = axa and x = xax, and (iii) a = axa with x invertible. Our main result: If a prime regular ring R contains an element a for which the number of solutions to (i), (ii), or (iii) is finite and greater than one, then R is a matrix ring over a finite field. Complete descriptions are given of those regular rings for which the number of solutions to (i), (ii), or (iii) is always one and those for which the number is always finite.

Mathematical Subject Classification
Primary: 16A30, 16A30
Secondary: 16A48
Received: 1 May 1978
Revised: 13 October 1979
Published: 1 April 1980
Thomas R. Savage