Definition:Multiplicative Inverse

Definition
Let $\left({A_F, \oplus}\right)$ be a unitary algebra whose unit is $1$ and whose zero is $0$.

Let $a \in A_F$ such that $a \ne 0$.

A multiplicative inverse of $a$ is an element $b \in A_F$ such that:
 * $a \oplus b = 1 = b \oplus a$

Comment
Although the term could equally well be applied to any inverse element in the context of a ring product in any ring, it is usually reserved for the context of a unitary algebra.