14
$\begingroup$

Suppose $u$ is an element of a ring with a right inverse. I'm trying to understand why the following are equivalent.

  • $u$ has at least two right inverses
  • $u$ is a left zero divisor
  • $u$ is not a unit

If $v$ and $w$ are distinct right inverse of $u$, then $u(v-w)=0$, but $v-w\neq 0$, so $u$ is a left zero divisor. It's also clear that if $u$ is a left zero divisor, it cannot be a unit (else I could cancel $u$ from $ub=0$ to see $b=0$).

I'm having a heck of a time seeing why $u$ is not a unit implies $u$ has at least two right inverses. I tried the contrapositive, but saw no good approach. What am I missing?

$\endgroup$
21
$\begingroup$

If $u$ has only one right inverse $v$, then $u(1-vu)=u-(uv)u=0$ hence $u(1-vu+v)=1$ and by uniqueness $1-vu+v=v$, so $1=vu$ and $v$ is a left inverse. Hence $u$ is a unit.

$\endgroup$
1
  • $\begingroup$ What if $u$ has no right inverse? $\endgroup$ Aug 5 '19 at 19:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.