associate

Two elements are associates if they differ only by a 'unit' factor — something invertible that does not change the essential content of a factorization, the way 3 and −3 are the same prime up to sign. Associates are the ring-theoretic version of 'the same up to an irrelevant twist.'

Formally, in a commutative ring R, two elements a and b are associates if a = u·b for some unit u (an element with a multiplicative inverse in R). This is an equivalence relation, and in an integral domain it can be restated symmetrically: a and b are associates exactly when a divides b and b divides a, equivalently when they generate the same principal ideal, (a) = (b).

The notion is essential for stating unique factorization cleanly: factorization into irreducibles is unique only 'up to order and associates,' since you can always shuffle units between factors. In Z the units are ±1, so the only associate of a is −a. In a field every nonzero element is a unit, so all nonzero elements are associates of one another, which is why factorization theory is vacuous in a field.