# Definition:Submultiplicative Function on Ring

## Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $f: R \to \R$ be a (real-valued) function on $R$.

$f$ is a submultiplicative function if and only if:

$\forall x, y \in R: \map f {x \circ y} \le \map f x \times \map f y$

That is, a submultiplicative function is one where the value of the product of two elements of $R$ is never greater than the product of the value of each.

## Also see

• Results about submultiplicative functions can be found here.