Three frameworks for a general theory of factorization

We discuss three different frameworks for a general theory of factorization in integral domains: τ-factorization, reduced τ-factorization and Γ-factorization. Let D be an integral domain, $${D^{\sharp}}$$ the non-zero, non-units of D, and τ a symmetric relation on $${D^{\sharp}}$$ . For $${a\in D^{\sharp}, a=\lambda a_{1}\cdots a_{n},\lambda}$$ a unit, $${a_{i}\in D^{\sharp}, n\geq1,}$$ and a i τa j for i ≠ j, is called a τ-factorization of a and we say a i is a τ-factor of a. For $${a,b\in D^{\sharp}, a\mid_{\tau}b}$$ if a is a τ-factor of b. Then $${a\in D^{\sharp}}$$ is a τ-atom if any τ-factorization of a has n = 1 and a is a τ-prime (resp., $${\mid_{\tau}}$$ -prime) if $${a\mid\lambda a_{1}\cdots a_{n}}$$ (resp., $${a\mid_{\tau}\lambda a_{1}\cdots a_{n}}$$ ), λ a1 . . . a n a τ-factorization, implies $${a\mid a_{i}}$$ (resp., $${a\mid_{\tau}a_{i}}$$ ) for some i. The theory of reduced τ-factorization is developed similarly, except here we restrict ourselves to reduced τ-factorizations, that is, τ-factorizations a1 . . . a n where the leading unit is omitted (or is 1). The theory of Γ-factorization is as follows. For $${a\in D^{\sharp},{\rm fact}(a)}$$ (resp., tfact(a)) is the set of (resp., trivial) factorizations of a, a = λ a1 . . . a n ,λ a unit, n ≥ 1 (resp,. n = 1) and $${{\rm fact}(D)=\cup_{a\in D^{\sharp}}{\rm fact}(a),{\rm tfact}(D)= \cup_{a\in D^{\sharp}}{\rm tfact}(a)}$$ . Let $${\Gamma\subseteq {\rm fact}(D)}$$ and $${\Gamma(a)=\Gamma\cap {\rm fact}(a)}$$ ; the set of Γ-factorizations of a. For $${a,b\in D^{\sharp}, a \mid_{\Gamma} b}$$ if some $${\lambda a_{1}\cdots a_{n} \in\Gamma(b)}$$ has a i = a for some i. We say a is a Γ-atom if $${\Gamma(a)\subseteq {\rm tfact}(a)}$$ and that a is a Γ-prime (resp., $${\mid_{\Gamma}}$$ -prime) if $${a\mid\lambda a_{1} \cdots a_{n}}$$ (resp., $${a\mid_{\Gamma}\lambda a_{1} \cdots a_{n}}$$ ) where $${\lambda a_{1}\cdots a_{n} \in\Gamma,}$$ then $${a\mid a_{i}}$$ (resp., $${a\mid_{\Gamma} a_{i}}$$ ) for some i.