Centre of a simple algebra is a field > How can one show that the centre of simple algebra is a field? I have tried it and proved that the inverse exists for every element of centre but cannot prove that inverse of every element belongs to centre. Please help me.

If $ab=ba$ and $a^{-1}$ exists, then $a^{-1}b=a^{-1}(ba)a^{-1}=a^{-1}(ab)a^{-1}=ba^{-1}$.