Proposition 1

*Let $A$ be an algebra of finite degree over a field $K$. If an element $a \in A$ is not a left (resp. right) divisor of zero in $A$, then it is invertible in $A$.

For the vector space $A$ over $K$ is of finite dimension, by hypothesis, and the linear mapping $x \mapsto ax$ (resp. $x \mapsto xa$) of $A$ into $A$ injective; it is therefore bijective (II, p. 298, Cor.) and hence (I, p. 16, Remark) $a$ is invertible in $A$.*