[Math] If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses.

Question

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Kaplansky's theorem of infinitely many right inverses in monoids?

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Let $A$ be a ring and $a\in A$ an element that has a right-inverse but does not have a left-inverse. Show that $a$ has infinitely many right-inverses.

Answer 1

Hint: Let $b$ be a right-inverse of $a$. For any $i \geq 0$, we define $b_i = (1-ba)a^i + b$. Show that if $a$ doesn't have a left-inverse, the $b_i$ are pairwise distinct right-inverses of $a$.