Sub-multiplicativity

In algebra, submultiplicativity and multiplicativity are properties of the fidelity of order of functions with respect to multiplication .

definition

Be a unitary ring . A mapping of into the nonnegative real numbers is called submultiplicative if the property applies to all ${\displaystyle R}$ ${\displaystyle f\colon R\to \mathbb {R} _{+}}$ ${\displaystyle R}$ ${\displaystyle a,b\in R}$

${\displaystyle f(a\cdot b)\leq f(a)\cdot f(b)}$

applies. If even the stricter requirement

${\displaystyle f(a\cdot b)=f(a)\cdot f(b)}$

is fulfilled, it is called multiplicative . ${\displaystyle f}$

Examples

If there is a unitary ring (which can also be a body , for example ), the requirement of submultiplicativity is one of the axioms for a pseudo amount . The requirement of multiplicativity is one of the axioms for an amount .

For further examples see also pseudo norm .