If $u,v$ and $auvb$ are units, then is $ba$ a unit?

Question

Suppose $R$ is a non-commutative ring

Let $a,b,u,v \in R$ such that $u,v$ are units and $auvb$ is a unit

Is it true that then $ba$ is a unit?

($x$ is a unit I mean that there exist $t$ such that $tx = xt = 1$)

I'm really stuck. thank you in advance

By the units being a group, this is equivalent to the same question without the $v$. — Torsten Schoeneberg, Sep 23 '19 at 17:08
If your ring is finite then yes since if either $a$ or $b$ is a zero divisor then $aub$ is also a zero divisor for every unit $u$. — Μάρκος Καραμέρης, Sep 23 '19 at 17:18

score 4 · Accepted Answer · answered Sep 23 '19 at 17:18

4

Take any ring in which $ab=1$ and $ba\neq 1$. This is a special case of $u=v=1$ for your hypotheses.

Then $ba$ is a nontrivial idempotent, hence not a unit.

answered Sep 23 '19 at 17:18

rschwieb

1 Answers1