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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!