the left inverse and right inverse in a ring

Question

It is well-known that if $R$ is an Artin ring,and $ab=1$ in $R$ where $a,b\in R$,then $ba=1a$.(This is not difficult)this is a very hot in Mathematics. If $AB = I$ then $BA = I$

It seems it is not right for arbitrary ring that if $ab=1$,then $ba=1$. Can someone helps to give an example.

Thanks in advance!

Answer 1

1. Staying at linear endomorphisms, this is no longer true for infinite dimensional spaces.
   Let e.g. $e_1,e_2,\dots$ be a basis, and consider $B:=e_k\mapsto e_{k+1}$ and $A:=e_k\mapsto e_{k-1}, \ e_1\mapsto 0$

2.Take any monoid $M$ that satisfies this, then consider its 'group ring' $\Bbb ZM$.