Factorial Domain Research Articles

Generating absolute prime ideals

A means of obtaining absolute prime ideals is presented. This process yields an example of a three-dimensional, local, non-Cohen-Macaulay factorial domain in characteristic zero.

Factorial domains and graded rings

(1989). Factorial domains and graded rings. Communications in Algebra: Vol. 17, No. 4, pp. 827-836.

Non-Catenary factorial domains

Non-Catenary factorial domains

On primary factorizations

We relate ideals in commutative rings which are products of primary ideals to ideals with primary decompositions. Invertible primary ideals are shown to have special properties. Sufficient conditions are given for a primary product ideal to have a unique product representation. A domain is weakly factorial if every non-unit is a product of primary elements. If R is weakly factorial, Pic( R)=0. A Noetherian weakly factorial domain R is factorial precisely when R is integrally closed. R[ X] is weakly factorial if and only if R is a weakly factorial GCD domain. Properties of weakly factorial GCD domains are discussed.

Locally almost factorial integral domains

The simpliest, and most important, class of Krull domains are the factorial domains (UFDs), i.e., those Krull domains with trivial divisor class group. Probably the next simpliest class of Krull domains are the Krull domains with torsion divisor class group, i.e., the almost factorial Krull domains [21]. Almost factorial Krull domains occur very frequently: examples include factorial domains, Krull domains with finite divisor class group, and hence any ring of integers of an algebraic number field. Moreover, Goldman [16, Corollary 21 has shown that any Dedekind domain R has an almost factorial overring A with the same unit group as R. The most important property of almost factorial Krull domains is that they are precisely the Krull domains R such that each subintersection of R is a localization of R. Other characterizations of almost factorial Krull domains may be found in [ 131 or [21]. Almost factorial domains have also been called semifactorial [ 191 and prefactorial. Recently there have been several papers on rings each of whose proper localizations or proper overrings satisfy certain ring-theoretic properties, for example [4,5, 8,9]. Following Fossum [ 13, p. 811, in [4] we defined

Knot modules and the elementary divisor theorem

We show that a knot module M is a cyclic module if and only if E 1( M)= Λ, all Steinitz-Fox-Smythe invariants of quotients of M by irreducible factors of Δ 0( M) are trivial, and certain unit class invariants describing the extensions in a composition series for M are trivial. (We give several counter-examples to illustrate the independence of these conditions.) We show also that a direct sum of cyclic modules (over any noetherian factorial domain) satisfies the Elementary Divisor Theorem if and only if all its elementary ideals are principal, and we replace the Dedekind condition in Levine's π-primary sequence realization theorem by a (weaker) projectivity condition.

Locally factorial integral domains

Following Fossum (9, p. 8 11, we define an integral domain R to be locally factorial if R, = R [ l/f] is factorial for each nonzero nonunit f E R. For example, any factorial integral domain or one-dimensional quasilocal integral domain is locally factorial. We show that a locally factorial integral domain R which is not quasilocal is a Krull domain, and R, is factorial for each maximal ideal M of R. We will be mainly interested in locally factorial Krull domains. In this case we can relate divisibility properties of R to group-theoretic properties of Cl(R), its divisor class group. The first examples of Dedekind domains which are locally factorial (PID), but not factorial (PID), are due to Claborn [6]. We know of no locally factorial Krull domains which are neither factorial nor Dedekind domains. Most of our results will be stated for arbitrary locally factorial Krull domains and subintersections of Krull domains. However, major emphasis will be towards Dedekind domains. In this case a factorial overring is just a PID, and each overring is a subintersection. In Section 2, we prove several basic results about locally factorial integral domains and give several examples of locally factorial Krull domains. Claborn’s examples actually satisfy the stronger property that each proper overring of R is factorial (PID). In Section 3, we study those Krull domains each of whose proper overrings is factorial. Such an integral domain which is not quasilocal is necessarily a Dedekind domain with cyclic divisor class 265

On 2-dimensional local rings with Artin's approximation property

AbstractExtending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

A note on locally factorial noetherian domains∗

A note on locally factorial noetherian domains^∗

Pure descent for the module of Zariski differentials

It will be shown that for any given pure extension A → B A \to B of noetherian k k -algebras, with k k being a field of characteristic zero, and for any prime ideal p ⊆ A \mathfrak {p} \subseteq A the Zariski-Lipman conjecture for A p {A_\mathfrak {p}} is solvable, if B B is a locally factorial domain for which the finite differential module is reflexive. We will also discuss an embedding property with respect to the module of Zariski differentials of A p {A_\mathfrak {p}} .

Graded factorial domains

Graded factorial domains

The Relationship among Measures of Cognitive Development, Learning Proficiency; Academic Achievement, and IQ for Seventh Grade, Low Socioeconomic Status Black Males

The degree of relationship among measures of learning proficiency, cognitive development (following Piaget), school achievement in reading, mathematics, and spelling, and IQ among 79 low socioeconomic status black males attending the seventh grade in an innercity school was examined. The following hypotheses were tested: (1) There would be a positive and significant relationship between measures of learning proficiency or cognitive growth and school achievement; (2) learning and cognitive variables would fall into a common factorial domain; and (3) the latter two variables would hold a reciprocal relationship within that domain. Hypothesis one was not supported, while two and three were supported. The latter two results supported Jensen’s two-level theory of learning. These findings and between-pupil differences in cognitive development were discussed.

Completely decomposable flat modules over locally factorial domains

Rank 1 1 flat modules are classified for a locally factorial noetherian domain by extending the concept of divisor. A direct sum of rank one flat modules in called completely decomposable. A summand of a completely decomposable module is a direct sum of homogeneous components but need not be completely decomposable.

Completely Decomposable Flat Modules over Locally Factorial Domains

Completely Decomposable Flat Modules over Locally Factorial Domains

An algorithm for rings of invariants

If R is a factorial domain and R o is a factorial subring containing all units of R, and if every prime element of R o remains prime in R, then being the fraction field of R o. This yields the first main theorem of invariant theory for several classical groups over an infinite factorial base ring.