Let R be a (noncommutative) ring with identity, such that there are elements u and v with u v = 1. Then the following are equivalent:
- u w = 0 for some nonzero w.
- u is not a unit.
- u has other right inverses.
Proof. It’s easy to see that 1 and 3 are equivalent — if v is an inverse, then so is v + w, and vice-versa — and clearly they imply condition 2. Showing that 2 implies the others isn’t as obvious, but it’s a nice one-liner formal trick.
In particular, note that u (1 – v u) = u – u v u = u – (u v) u = u – u = 0, so either v u = 1 and hence u is a unit, or w = 1 – vu is a nonzero element of R such that u w = 0.
In fact, by a theorem due to Kaplansky there are infinitely many such inverses if there are two, which we obtain simply by taking . To see these are distinct, note that if
for some n < m, then multiplying through by on the right gives
i.e. u is actually a unit, contradicting the hypothesis.