Exercise 3, § 2

Let $A$ be an algebra of finite degree over a commutative field $K$, where $A$ is not assumed unital. Show that if $a \in A$ is cancellable on the left, then there exists $e \in A$ such that $ex = x$ for all $x \in A$ and there exists $b \in A$ with $ab = e$. If $a$ is also cancellable on the right, $e$ is the unit element of $A$ and $a$ is invertible in $A$. The case of commutative $A$.