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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?

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1 Answer 1

up vote 12 down vote accepted

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.

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Thanks. (chars) –  kathleen Jun 9 '12 at 10:00
    
You are welcome. –  Davide Giraudo Jun 9 '12 at 13:14

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