Talk:Division Algebra has No Zero Divisors

Algebra must be finite dimensional over the field

If the algebra has no zero divisors and is fd over the field, then it is a division algebra.

To prove this equivalence, we need the algebra $A$ to be finite dimensional. In this case, the left and right multiplications by an arbitrary non zero element $t$ $L_t,R_t:A\to A$ have zero kernel since there are no zero divisors. By applying the first isomorphism theorem (here we require $A$ is finite dimensional) $Im(R_t)=Im(L_t) =A$. The preimages of this functions give us the elements we want.