Definition: Factorial Ring, Generalization of Factorization

An integral domain $R$ is called a factorial ring, if every $a\in R\setminus \{0\}$ has the factorization $$a=\prod_{i=1}^r p_i^{e_i}$$ of irreducible elements $p_i$ and positive integer exponents $e_i > 0,$ which is unique except of the order of the elements $p_i$ and the associates of all $p_i.$

Definitions: 1
Lemmas: 2

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  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
  2. Koch, H.; Pieper, H.: "Zahlentheorie - Ausgewählte Methoden und Ergebnisse", Studienbücherei, 1976