Prove that if $R$ is a non-commutative ring with $1$ and if $a,b \in R$ and if $ab =1 $ but $ba \neq 1$ then $R$ is infinite Prove that if $R$ is a non-commutative ring with $1$ and if $a,b \in R$ and if  $ab =1 $ but $ba \neq 1$ then $R$ is infinite
I'm not sure about this one.
I noticed that $ba$ is an idempotent, and thus $(1-ba)$ is idenmpotent... So far I have 5 elements wooh! Only need $\infty$ more! Haha but seriously i'm pretty confused, insight appreciated!

Hint: show that if $R$ is finite, then $a^n = 1$ for some natural number $n$. Now compute $b$.
