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

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.

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The title should be more helpful –  Shu Xiao Li Mar 13 '13 at 18:08
Please formulate this question more clearly. –  Alexander Gruber Mar 13 '13 at 18:08
Hmm, I duped it as the wrong question. I'm pretty sure this has already been asked though. –  rschwieb Mar 13 '13 at 18:10
The question is almost certainly "If $a$ is left invertible but not right invertible, show $a$ has infinitely many distinct left inverses", a classic result. –  rschwieb Mar 13 '13 at 18:11
So I understand this question was closed because of the unbelievable language it was "written" in? Because it is not the duplicate of Kaplansky's theorem, at least not directly. –  DonAntonio Mar 14 '13 at 0:15
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## marked as duplicate by Alexander Gruber♦, rschwieb, Seirios, Davide Giraudo, ArkamisMar 13 '13 at 18:35

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 $a$.