Abstract
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.
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References
Anderson, D.D.: Non-atomic factorization in integral domains. In: Chapman, S. (ed.): Arithmetical Properties of Commutative Rings and Monoids. Lecture Notes in Pure and Applied Math., vol. 241, pp. 1–21. CRC Press, Boca Raton (2005)
Anderson D.D., Anderson D.F., Zafrullah M.: Factorization in integral domains. J. Pure Appl. Algebra 69, 1–19 (1990)
Anderson D.D., Frazier A.: On a general theory of factorization in integral domains. Rocky Mt. J. Math 41, 663–705 (2011)
Frazier, A.M.: Generalized factorizations in integral domains, PhD Thesis, The University of Iowa (2006)
Geroldinger, A.; Halter-Koch, F.: Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278. Taylor and Francis, CRC Press, Boca Raton (2006)
Gilmer, R.: Multiplicative Ideal Theory. Queen’s Papers in Pure and Appl. Math., vol. 90, Queen’s University, Kingston, Ontario (1992)
Hamon, S.M.: Some topics in τ-factorizations, PhD Thesis, The University of Iowa (2007)
Juett, J.: Two counterexamples in abstract factorization. Rocky Mt. J. Math. (in press)
Juett, J.: Some topics in Γ-factorizations in integral domains, PhD Thesis, The University of Iowa (in preparation)
McAdam, S.; Swan, R.G.: Unique comaximal factorization. J. Algebra 276, 180–192 (2004)
Ortiz-Albino, R.M.: On generalized nonatomic factorizations, PhD Thesis, The University of Iowa (2008)
Reinkoester, J.: Relative primeness, PhD Thesis, The University of Iowa (2010)
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Anderson, D.D., Ortiz-Albino, R.M. Three frameworks for a general theory of factorization. Arab. J. Math. 1, 1–16 (2012). https://doi.org/10.1007/s40065-012-0012-7
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DOI: https://doi.org/10.1007/s40065-012-0012-7