A" υ-Operation Free" Approach to Pr¨ ufer υ-Multiplication Domains

The so-called Prüferυ-multiplication domains (PυMDs) are usually defined as domains whose finitely generated nonzero ideals aret-invertible. These domains generalize Prüfer domains and Krull domains. The PυMDs are relatively obscure compared to their very well-known special cases. One of the reasons could be that the study of PυMDs uses the jargon of star operations, such as theυ-operation and thet-operation. In this paper, we provide characterizations of and basic results on PυMDs and related notions without star operations

On vv--domains and star operations

Let $\ast$ be a star operation on an integral domain $D$. Let \f(D) be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\ast$--Pr\"ufer (respectively, $(\ast, v)$--Pr\"ufer) domain if $(FF^{-1})^{\ast}=D$ (respectively, $(F^vF^{-1})^{\ast}=D$) for all F\in \f(D). We establish that $\ast$--Pr\"ufer domains (and $(\ast, v)$--Pr\"ufer domains) for various star operations $\ast$ span a major portion of the known generalizations of Pr\"{u}fer domains inside the class of $v$--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Pr\"ufer domain, as a model, and we show which statements of that theorem on Pr\"ufer domains can be generalized in a natural way and proved for $\ast$--Pr\"ufer domains, and which cannot be. We also show that in a $\ast$--Pr\"ufer domain, each pair of $\ast$-invertible $\ast$-ideals admits a GCD in the set of $\ast$-invertible $\ast$-ideals, obtaining a remarkable generalization of a property holding for the "classical" class of Pr\"ufer $v$--multiplication domains. We also link $D$ being $\ast$--Pr\"ufer (or $(\ast, v)$--Pr\"ufer) with the group Inv$^{\ast}(D)$ of $\ast$-invertible $\ast$-ideals (under $\ast$-multiplication) being lattice-ordered

Unique factorization and related topics

This work can he split into two parts. In the first part we generalize the concept of Unique Factorization by viewing Unique Factorization Domains as integral domains, non zero non units of which can he expressed uniquely (up to associates and order) as products of finitely many mutually co-prime associates of prime powers. Our working rule consists of taking a subset Q of the set P of all properties of a general prime power and investigating integral domains, whose non zero non units are expressible uniquely as proproducts of finitely many non units satisfying the properties in Q. For example we take Q consisting of only one property: of any two factors of a prime power one divides the other and call a non unit x rigid if for each h,k dividing x one divides the other. We find that in a Highest Common Factor domain a product of finitely many rigid elements is expressible uniquely as the product of mutually co-prime rigid elements. And a Highest Common Factor domain with the set of non zeros generated by rigid elements and units is the resulting generalization of a Unique Factorization Domain. We consider three different Q's which for suitable integral domains give distinct generalizations of Unique Factorization domains. In each case we provide examples to prove their existence discuss their points of difference with UFD's and study their behaviour under localization and adjunction of indeterminates. We also study these integral domains in terms of the valuations of their fields of fractions and show that these integral domains are generalizations of Krulldomains. The second part is mainly a study of ideal transforms in generalized Krull domains and some of the results are generalizations of results known for Krull domains.<p