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

so

i.e. *u* is actually a unit, contradicting the hypothesis.

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