Rings with a theory of greatest common divisors. (English) Zbl 0901.13001

We quote the author’s words: “In this paper we introduce the concept of a ring with gcd-theory to generalize the notion of a gcd-domain. We call a domain \(D\) ring with gcd-theory if there is a homomorphism \((\cdot)\) from \(D^*\) into a gcd-monoid \({\mathcal G}^+\) such that the divisibility properties of \(D\) are faithfully transferred to \({\mathcal G}^+\) and two elements of \({\mathcal G}^+\) are equal if they divide the same elements \((a)\), \(a\in D^*\). By means of *-ideal theory we study the fundamental properties of rings with gcd-theory. We show that a ring \(D\) with gcd-theory is simply a \(\nu\)-domain. We examine the behaviour of rings with gcd-theory when passing on to the integral closure in algebraic field extensions. Finally, we take a closer look at special cases of gcd-theories, namely to rings with gcd-theory of finite type, i.e. rings with gcd-theory in which any element of \({\mathcal G}^+\) can be written as the greatest common divisor of a finite number of elements \((a)\), \(a\in D^*\), and to the well-known rings with divisor theory”.


13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F07 Euclidean rings and generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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[1] Anderson, D. D./Mott, J./Zafrullah, M.: Some Quotient Based Statements in Multiplicative Ideal Theory. – Boll. Unione Mat. Ital., VII. Ser.,B 3 (1989), 456–476. · Zbl 0676.13003
[2] Aubert, Karl Egil: Divisors of finite character. – Ann. Mat. Pura Appl.38 (1983), 327–360. · Zbl 0533.20034
[3] Becker, Ulrich: Kronecker-Divisoren und ggT-Theorie. Dipl. thesis. Math. Inst. der Georg-August-Universität Göttingen 1992 (unpubl.).
[4] Borewicz, Senon I./Šafarevič, Igor R.: Zahlentheorie. – Basel/Stuttgart: Birkhäuser 1966.
[5] Bourbaki, Nicolas: Commutative Algebra. Chapters 1–7. – Berlin u.a.: Springer 1989. · Zbl 0666.13001
[6] Geroldinger, A./Močkoř, J.: Quasi-divisor theories and generalizations of Krull domains. – J. Pure Appl. Algebra102 (1995), 289–311. · Zbl 0853.13012
[7] Gilmer, Robert: Multiplicative Ideal Theory. – Kingston (Ontario): Queen’s Papers 1992. · Zbl 0804.13001
[8] Jaffard, Paul: Systèmes Idéaux. – Paris: Dunod 1960.
[9] Kang, B. G.: Prüferv-Multiplication Domains and the RingR[X] Nu . – J. Algebra123 (1989), 151–170. · Zbl 0668.13002
[10] Koch, Helmut: Zur Begründung der Arithmetik in algebraischen Zahl und Funktionenkörpern. – Wiss. Z. Humboldt-Univ. Berlin. Math.-naturw. R.15 (1966), 187–189. · Zbl 0148.27704
[11] Koch, H.: Number Theory II. Algebraic Number Theory. (Encyclopedia of Mathematical Sciences 62) – Berlin u.a.: Springer 1992.
[12] Kronecker, Leopold: Grundzüge einer arithmetischen Theorie der algebraischen Größen. – J. Reine Angew. Math.92 (1882), 1–122. · JFM 14.0038.02
[13] Krull, Wolfgang: Beiträge zur Arithmetik kommutativer Integritätsbereiche. II.v-Ideale und vollständig ganz abgeschlossene Integritätsbereiche. – Math. Z.41 (1936), 665–679. · Zbl 0015.24501
[14] Lucius, Friedemann: Ringe mit einer Theorie des größten gemeinsamen Teilers. – Math. Gottingensis, Heft7 (1997), 1–70. · Zbl 0901.13002
[15] Mott, Joe L./Zafrullah, Muhammad: On Prüferv-multiplication domains. – Manuscr. Math.26 (1981), 1–26. · Zbl 0477.13007
[16] Močkoř, Jiří: Groups of Divisibility. – Dodrecht u.a.: Reidel 1983.
[17] Močkoř, Jiři/Kontolatou, Angeliki: Groups with Quasi-divisors Theory. – Comment. Math. Univ. St. Pauli42 (1993), 23–36. · Zbl 0794.06015
[18] Prüfer, Heinz: Untersuchungen über Teilbarkeitseigenschaften in Körpern. – J. Reine Angew. Math.168 (1932), 1–36. · Zbl 0004.34001
[19] Querre, Julien: Idéaux divisoriels d’un anneau de polynômes. – J. Algebra64 (1980), 270–284. · Zbl 0441.13012
[20] Skula, Ladislav: Divisorentheorie einer Halbgruppe. – Math. Z.114 (1970), 113–120. · Zbl 0185.04804
[21] Swan, R. G.:n-generator Ideals in Prüfer domains. – Pac. J. Math.111 (1984), 433–446. · Zbl 0546.13008
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