In algebra, an absolute value is a function that generalizes the usual absolute value. 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:

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 the square of the classical absolute value is not, as it does not fulfill the triangular inequality.

An absolute value induces a metric (and thus a topology) on D by setting ${\displaystyle d(x,y)=|x-y|.}$

The trivial absolute value is the absolute value with |x| = 0 when x = 0 and |x| = 1 otherwise. Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a finite field because any non-zero element can be raised to some power to yield 1.

If an absolute value satisfies the stronger property |x + y| ≤ max(|x|, |y|) for all x and y, then |x| is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value.

If |x|₁ and |x|₂ are two absolute values on the same integral domain D, then the two absolute values are equivalent if |x|₁ < 1 if and only if |x|₂ < 1 for all x. If two nontrivial absolute values are equivalent, then for some exponent e we have |x|₁^e = |x|₂ for all x. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it violates the rule |x+y| ≤ |x|+|y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place.

Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p (see the third example above.)