On density theorems for rings of Krull type with zero divisors

Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap R(Pi). We denote by Gi the value group of the valuation associated with R(Pi). We prove that there is an order homomorphism from I(R) into the cardinal direct sum \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.

Anahtar Kelimeler:

Krull ring, ring of Krull type, valuation Marot ring

On density theorems for rings of Krull type with zero divisors

Keywords:

Krull ring, ring of Krull type, valuation Marot ring,

PDF

___

Atiyah MF, MacDonald IG. Introduction to Commutative Algebra. Boston, MA, USA: Addison-Wesley Publishing Company, 1969.
Brewer J, Klingler L. The ordered group of invertible ideals of a Pr¨ ufer domain of finite character. Commun Algebra 2005; 33: 4197–4203.
Fuchs L, Salce L. Modules over Non-Noetherian Domains. Mathematical Surveys and Monographs. New York, NY, USA: Marcel Dekker, 2000.
Gilmer R. Multiplicative Ideal Theory. New York, NY, USA: Marcel Dekker, 1976.
Gilmer R, Huckaba JA. ∆ Rings. J Algebra 1974; 28: 414–432
Glaz S. Controlling the zero divisors of a commutative ring. Lect Notes Pure Appl 2002; 231: 191–212.
Halter-Koch F. A characterization of Krull rings with zero divisors. Archivum Mathematicum 1993; 29: 119–122. Huckaba JA. Commutative Rings with Zero Divisors. New York, NY, USA: Marcel Dekker, 1988.
Kelly PH, Larsen MD. Valuation rings with zero divisors. P Am Math Soc 1971; 30; 3: 426–430.
Lantz DC, Martin MB. Strongly two-generated ideals. Commun Algebra 1988; 16; 9: 1759–1777.
Manis M. Valuations on a commutative ring. P Am Math Soc 1969; 20: 193–198.
Marot J. Une g´ en´ eralisation de la notion danneau de valuation. C R Acad Sc Paris 1969; 268: A1451–A1454.