\begin{definition}[Definition:Completely Additive Function]
Let $\left({R, +, \times}\right)$ be a ring.
Let $f: R \to R$ be a mapping on $R$.
Then $f$ is described as '''completely additive''' {{iff}}:
:$\forall m, n \in R: f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$
That is, a '''completely additive function''' is one where the value of a product of two numbers equals the sum of the value of each one individually.
\end{definition}