prove that $ab=1$ implies $ba=1$.

Question

I have a doubt how to prove:
If the product of any pair of non-zero elements of $R$ is non-zero , prove that $ab=1$ implies $ba=1$.

how shall I make use of the fact : product of any pair of non-zero elements of $R$ is non-zero i.e no zero divisors exist to prove this...

Answer 1

$$ ab=1 \implies aba=a \implies a(ba-1)=0 \implies ba=1 $$