Definition:Complete Factorization

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Let $\struct {D, +, \circ}$ be an integral domain.

Let $x$ be a non-zero non-unit element of $D$.

A complete factorization of $x$ in $D$ is a tidy factorization:

$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$

such that:

$u$ is a unit of $D$
all of $y_1, y_2, \ldots, y_n$ are irreducible in $D$.

Linguistic Note

The spelling factorization is the US English version.

The UK English spelling is factorisation.