## Definition

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 if and only if:

$\forall m, n \in R: f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$

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.