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.
Asked
Active
Viewed 2,214 times
6
-
The title should be more helpful – NECing 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 00:15
-
@DonAntonio Yeah, it's not a duplicate of the question, but the question text contains Kaplansky's argument (identical to azimut's below). – rschwieb Mar 14 '13 at 00:28
1 Answers
4
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 $a$.
azimut
- 22,696