Muhammad Zafrullah

Factorization in integral domains

Abstract In this paper, we study several factorization properties in an integral domain which are weaker than unique factorization. We study how these properties behave under localization and directed unions.

The construction D + XDS[X]

If D is a commutative integral domain and S is a multiplicative system in D, then Tfs) = D + XD,[X] is the subring of the polynomial ring D,[X] con- sisting of those polynomials with constant term in D. In the special case where S = D* = D\(O), we omit the superscript and let T denote the ring D + XK[XJ, where K is the quotient field of D. Since Tfs) is the direct limit of the rings D[X/s], where s E S, we can conclude that many properties hold in T@) because these properties are preserved by taking polynomial ring extensions and direct limits. Moreover, the ring Tcs) is the symmetric algebra S,(D,) of D, considered as a D-module. In addition, Ds[Xj is a quotient ring of Tts) with respect to S; in fact, in the terminology of [lo], Tfs) is the composite of D and D,[iYj over the ideal XDJX]. (The most familiar of the composite constructions is the so-called D + M construction [l], where generally M is the maximal ideal of a valuation ring.) The ring T ts), therefore, provides a test case for many questions about direct limits, symmetric algebras, and composites. The state of our knowledge of T is considerably more advanced than that of VJ; generally speaking, we often show that a property holds in T if and only if it holds in D. In other cases we show that Tcs) does not have a given property if D, # K. For example, if T(S) is a Priifer domain, then D,[xJ is a Prtifer domain and D, is therefore equal to K. We show that T is Priifer (Bezout) if and only if D is Prtifer (Bezout). Yet Tts) is a GCD-domain if D is a GCD- domain and the greatest common divisor of d and X exists in

On finite conductor domains

An integral domain D is a FC domain if for all a, b in D, aD∩bD is finitely generated. Using a set of very general and useful lemmas, we show that an integrally closed FC domain is a Prüfer v-multiplication domain (PVMD). We use this result to improve some results which were originally proved for integrally closed FC domains (or for coherent domains) to results on PVMDs. Finally we provide examples of integrally closed integral domains which are not FC domains.

Almost Bézout domains

An integral domain R is said to be an almost Bezout domain (respectively, almost GCD-domain) if for x, y ϵ R − {0}, there exists an n with (xn, yn) (respectively, (xn, yn)v) principal. In this paper we continue the investigation of AGCD-domains begun by the second author and introduce the notion of an almost Bezout domain. We show that R is an almost Bezout domain if and only if R, the integral closure of R, is a Prufer domain with torsion class group and for every x ϵ R, there exists an n with xsn ϵ R.

A general theory of almost factoriality

Let R be a commutative integral domain with 1. The non-zero elements a,b of R may be calledv-coprime if aR∩bR=abR. A Krull domain is calledalmost factorial if for all f,g in R there is n∈N such that fnR∩gnR is principal. From this it is easy to establish that if R is almost factorial then for all x in R there is n∈N such that xn=p1p2...pr where pi are mutually v-coprime primary elements and that this expression is unique. In this article we drop the requirement that R be Krull and replace the primary elements by elements calledprime blocks and develope a theory of almost factoriality, a special case of which is the theory of almost factorial Krull domains.

Putting T-Invertibility to Use

This article gives a survey of how the notion of t-invertibility has, in recent years, been used to develop new concepts that enhance our understanding of the multiplicative structure of commutative integral domains. The concept of t-invertibility arises in the context of star operations. However, in general terms a (fractional) ideal A, of an integral domain D, is t-invertible if there is a finitely generated (fractional) ideal F ⊆ A and a finitely generated fractional ideal G ⊆ A -l such that (FG)-1 = D. In a more specialized context the notion of t-invertibility has to do with the t-operation which is one of the so called star operations. There seems to be no book other than Gilmer’s [Gil] that treats star operations purely from a ring theoretic view point. But a lot has changed since Gilmer’s book was published. So I have devoted a part of section 1. to an introduction to star operations, *-invertibility in general, and t-invertibility in particular.

The D + XDS[X] construction from GCD-domains

Abstract Let D be an integral domain, S a multiplicative set in D and X an indeterminate over D . We show that if D is a GCD-domain, then the behaviour of D ( S ) = { a 0 + Σ a i X i | a 0 ϵ D , a i ϵ D S } = D + XD S [ X ] depends upon the relationship between S and the prime ideals P of D such that D P is a valuation domain. We use this study to construct locally GCD-domains which are not GCD-domains. These domains have, each, at least one prime t-ideal P such that PD P is not a t-ideal. We also give an example of an ideal A with A⊊A t ⊊A v ⊊D. The D ( S ) construction is also used to construct, from lattice ordered groups, Riesz groups which are not lattice ordered.

t-linked extensions, the t-class group, and Nagata's theorem

Abstract Let A be a subring of the integral domain B . Then B is said to be t-linked over A if for each finitely generated ideal I of A with I -1 = A , we have ( IB ) -1 = B . If A and B are Krull domains, this condition is equivalent to PDE. We show that if B is t-linked over A , then the map I →( IB ) t gives a homomorphism from the group of t-invertible t-ideals of A to the group of t-invertible t-ideals of B and hence a homomorphism Cl t ( A )→Cl t ( B ) of the t-class groups. Conditions are given for these maps to be surjective which extend Nagatas Theorem for Krull domains to a much larger class of domains including, e.g., Noetherian domains each of whose grade-one prime ideals has height one.

Weakly factorial domains and groups of divisibility

An integral domain R is said to be weakly factorial if every nonunit of R is a product of primary elements. We give several conditions equivalent to R being weakly factorial. For example, we show that the following conditions are equivalent: (1) R is weakly factorial; (2) every convex directed subgroup of the group of divisibility of R is a cardinal summand; (3) if P is a prime ideal of R minimal over a proper principal ideal (x), then P has height one and (x)p n R is principal; (4) R = nRp , where the intersection runs over the height-one primes of R, is locally finite, and the t-class group of R is trivial. Throughout this note, R will be a commutative integral domain with identity having quotient field K. Suppose that R is a UFD. If S is a saturated multiplicatively closed subset of R, then S = {..Ap1 ... p, I. a unit of R, n > 0, each (pi) E Y} for some subset Y C X(1), where X(1) is the set of height-one prime ideals of R. Conversely, every such subset Y C X(1) determines a saturated multiplicatively closed subset of R. If we set T = {AP1 Pnl IA a unit of R, n > 0, each (pi) E X(1) Y}, then every nonzero element r of R may be written uniquely up to units in the form r = st where s E S and t E T. Stated in terms of the group of divisibility G(R) of R, (S) is a cardinal summand of G(R). In fact, (S) EC (T) = G(R). (See the next paragraph for definitions.) Here G(R) is a cardinal sum of copies of Z, one for each height-one prime, and (S) is the sum of the cyclic summands corresponding to the primes from Y. The purpose of this paper is to characterize the integral domains with this property. We show that an integral domain R has the property that every convex directed subgroup of G(R) is a cardinal summand of G(R) if and only if every nonunit of R is a product of primary elements, that is, R is weakly factorial. Let R be an integral domain with quotient field K. The group of divisibility of R is the group G(R) = K*/U(R), where K* is the multiplicative group of K and U(R) is the group units of R. G(R) is partially ordered Received by the editors July 1, 1989 and, in revised form, November 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 3A05, 1 3A 17, 1 3F99, 06F20. *This paper was written while the second author was visiting the University of Iowa. (D 1990 American Mathematical Society 0002-9939/90

WELL BEHAVED PRIME t-IDEALS

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