Associative Algebra has Multiplicative Inverses iff Unitary Division Algebra

Theorem
Let $\left({A_R, \oplus}\right)$ be an associative algebra over the ring $A_R$.

Then $\left({A_R, \oplus}\right)$ has a multiplicative inverse for every $a \in A_R$ iff $\left({A_R, \oplus}\right)$ is a unitary division algebra.