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.
