unique factorization domain

《代数学引论》 · 第 140 页

如果 $R$ 满足下列两条件，则 $R$ 叫做一个唯一因子分解整环，也叫高斯整环：……

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《A Course in Finite Group Representation Theory》 · 第 46 页 · Section 2.4「Prime Factorization」，Proposition and Definition 10

For an integral domain $R$, the following conditions are equivalent: (i) Every element $a\in R-(R^*\cup\{0\})$ can be uniquely written, up to associatedness and order, as a product of irreducible elements. (ii) Every element $a\in R-(R^*\cup\{0\})$ is a product of prime elements. An integral domain $R$ satisfying the equivalent conditions (i) and (ii) is called a factorial ring or a unique factorization domain.