[[Product ideal]]
# Unique factorization of ideals

A [[commutative ring]] $R$ admits **unique factorization of ideals** or **UFI** iff every nonzero proper ideal $I \triangleleft R$ may be written uniquely (up to ordering) as the [[Product ideal|product]] of [[Prime ideal|prime ideals]] #m/def/ring 
i.e. there is a bijection between ideals $\mathcal{I}$ and multisets $\mathbb{N}_{0}^\mathcal{P}$ of prime ideals.

## Rings with UFI

- [[A Dedekind domain admits UFI]]

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#state/tidy | #lang/en | #SemBr