Absolute value (algebra)

In algebra, an absolute value^[a] is a function that generalizes the usual absolute value.^[1] More precisely, if D is a field or (more generally) an integral domain, an absolute value on D is a function, commonly denoted ${\displaystyle |x|,}$ from D to the real numbers satisfying:

	•	${\displaystyle \left|x\right|\geq 0}$		(non-negativity)
	•	${\displaystyle \left|x\right|=0}$ if and only if ${\displaystyle x=0}$		(positive definiteness)
	•	${\displaystyle \left|xy\right|=\left|x\right|\left|y\right|}$		(multiplicativity)
	•	${\displaystyle \left|x+y\right|\leq \left|x\right|+\left|y\right|}$		(triangle inequality)

It follows from the axioms that ${\displaystyle |1|=1,}$ ${\displaystyle |-1|=1,}$ and ${\displaystyle |-x|=|x|}$ for every ⁠${\displaystyle x}$⁠. Furthermore, for every positive integer n, ${\displaystyle |n|\leq n,}$ where the leftmost n denotes the sum of n summands equal to the identity element of D.

The classical absolute value and its square root are examples of absolute values, but not the square of the classical absolute value, which does not fulfill the triangular inequality.

An absolute value such that ${\displaystyle |x+y|\leq \max(|x|,|y|)}$ is an ultrametric absolute value.

An absolute value induces a metric (and thus a topology) by ${\displaystyle d(f,g)=|f-g|.}$

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