Integrally closed and componentwise linear ideals

In a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is Cohen–Macaulay. In higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings R of arbitrary dimension. We identify a class of integrally closed ideals, the Goto-class G^*{\mathcal {G}^*}, which is closed under product and it has a suitable unique factorization property. Ideals in G^*{\mathcal {G}^*} have a Cohen–Macaulay associated graded ring if either they are monomial or dim R ≤ 3. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.     

Factorization of monic polynomials

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensel's Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.

     

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.     

Generalized comaximal factorization of ideals

We generalize the notion of comaximal factorization of ring ideals to the language of weak ideal systems on monoids and prove several results generalizing and extending previous work. We also develop some topological methods for dealing with comaximal factorization and some related finitary weak ideal system problems.     

On the Ascent of Properties Related to Unique Factorization Domains

In this article, we develop equivalent conditions for a certain class of monoidal transform to inherit either the property of being a completely integrally closed domain that satisfies the ascending chain condition on principal ideals, the property of being a Mori domain, the property of being a Krull domain, or the property of being a unique factorization domain, respectively. Such a class of monoidal transform is given in terms of an (analytically) independent set that forms a prime ideal in the base domain. Characterizations are provided illustrating the necessity of the “prime ideal” hypothesis when the base domain is a Noetherian unique factorization domain.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Properties of the normset relating to the class group

In a recent paper by the author the normset and its multiplicative structure was studied. In that paper it was shown that under certain conditions (including Galois) that a normset has unique factorization if and only if its corresponding ring of integers has unique factorization. In this paper we shall examine some of the properties of a normset and describe what it says about the class group of the corresponding ring of integers.

     

Rigid Rings and Makar-Limanov Techniques

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.     

UNIQUE FACTORIZATION AND BIRTH OF ALMOST PRIMES

ABSTRACT

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.     

Local and global methods in representations of Hecke algebras

This paper aims at developing a “local-global” approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the “good” prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.     

Complete ideals and singularities of space curves

We consider complete ideals supported on finite sequences of infinitely near points, in regular local rings with dimensions greater than two. We study properties of factorizations in Lipman special *-simple complete ideals. We relate it to a type of proximity, linear proximity, of the points, and give conditions in order to have unique factorization. Several examples are presented. Received: 2 February 2000 / in final form: 14 March 2001 / Published online: 18 January 2002     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

Commutative ideal theory without finiteness conditions: Completely irreducible ideals

An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.

     

關於素性環

<正> §1.本文的目的是在對於所謂素性環(Primal Ring)作一些探討.這裹的環都是指着有么元無零因子的可換環.我們以R表這樣一個環,1就是R的么元,大寫字母A,B,C,P,……表R的真理想子環,小寫字母a,b,c,x,y等表R的元.符號Ax~(-1)表示R中一切能使xy∈A的y所組成的集.容易證明Ax~(-1)是一個理想子環,並且Ax~(-1)A.如果Ax~(-1)A,則說x不素於A,否則說x素於A.這樣一來,A是素理想子環的充要條件就是R中凡不屬A的元都素於A.     

Decomposing finitely generated torsion modules

Let R be a commutative ring and let F be a filter of ideals of R. We give three characterizations of the rings with the property that all finitely generated F pretorsion R modules decompose into a direct sum of cyclic R submodules. One of these characterizations involves decomposing the filter F into a product of subfilters. This filter decomposition is shown to be unique, and is used to characterize several other ring properties relative to the filter of ideals.     

The radical-annihilator monoid of a ring

Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.     

Closed operator ideals and interpolation

We consider closed operator ideals, which mean operator ideals A whose components A(E, F) are closed subspaces of the space L(E, F). Using interpolation techniques, we obtain general results on products of closed ideals. Furthermore, we investigate which closed ideals A possess the factorization property, i.e., each operator of A factors through a space with the related property “A”. Applications of these results yield the answer to some open questions in ideal theory.     

On Fragility of Generalizations of Factoriality

Any class of domains, in particular a class of domains that arises from generalizations of factoriality, invites questions about its stability under the standard operations. One of these generalizations of factoriality is the one that requires that every nonzero element be contained in only finitely many principal prime ideals of height one. We use this property to settle all the open cases in the literature on stability of generalizations of factoriality under the standard ring extensions. The paper provides a compendium on the stability, under ring extensions, of all the known generalizations of factoriality. We also use stability properties of factorization in extensions of valuation domains to give a new characterization of discrete valuation domains.     

The enumeration of irreducible combinatorial objects

A unique factorization theory for labelled combinatorial objects is developed and applied to enumerate several families of objects, including certain families of set partitions, permutations, graphs, and collections of subintervals of [1, n]. The theory involves a notion of irreducibility with respect to set partitions and the enumeration formulas that arise result from a generalization of the well-known “exponential formula.”     

An example of an atomic pullback without the ACCP

We find necessary and sufficient conditions on a pullback diagram in order that every nonzero nonunit in its pullback ring admits a finite factorization into irreducible elements. As a result, we can describe a method of easily producing atomic domains that do not satisfy the ascending chain condition on principal ideals.