Commutative Regular Ring Research Articles

Artinian* Modules

In this paper, we introduce and study Artinian* modules as a generalization of Artinian modules. We transfer several results of Artinian modules to Artinian* modules. We also provide several characterizations of this new class of modules. Furthermore, we investigate the existence of secondary representation for Artinian* modules over commutative regular rings. Finally, we characterize Artinian* modules in the amalgamated module construction.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Symmetry on zero and idempotents

Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, for implies and for implies respectively, where S is a subset of a ring R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial over a symmetric ring is a unit then a 0 is a unit and ai is nilpotent for all based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

Pg-Extensions and p-Extensions with applications to C(X)

Essential extensions and p-extensions have been studied for commutative rings with identity in various papers, such as [P. Bhattacharjee, M. L. Knox and W. Wm. McGovern, p-Extensions, Proceedings for the OSU-Denison Conference, AMS series Contemporary Mathematics Series (to appear); p-Embeddings, Topology Appl. 160(13) (2013) 1566–1576; R. M. Raphael, Algebraic Extensions of Commutative Regular Rings, Canad. J. Math. 22(6) (1970) 1133–1155]. The present paper applies these concepts to certain subrings of C(X). Moreover, the paper introduces a new ring extension, called a pg-extension, and determines its relation to both essential extension and p-extension. It turns out that the pg-extension R ↪ S induces a well-defined contraction map between principal ideals [Formula: see text] and [Formula: see text].

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X n h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q 1 ⊕ Q 2 ⊕ Q 3 such that Q 1 is a ring satisfying S 2n−2, the standard identity of degree 2n − 2, Q 2 ≅ M n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x q = x for all x ∈ E, and Q 3 is a both faithful S 2n−2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.

Socle chains of abelian regular semiartinian rings

Let R be an abelian regular and semiartinian ring with socle chain (Sα∣α≤σ). If λα denotes the rank of the semisimple module Sα+1/Sα, for every α<σ, then the dimension sequence(λα∣α<σ) is an invariant for R. By applying classical results of combinatorial set theory we prove necessary conditions satisfied by this invariant. On the other hand, we present constructions of commutative regular semiartinian rings with given ranks of slices of socle chain. In some particular cases we prove a necessary and sufficient condition under which there exists an abelian regular semiartinian ring with given ranks of slices.

ON THE FINITENESS OF BASS NUMBERS OF LOCAL COHOMOLOGY MODULES

Let R be a commutative regular ring of dimension d, I be an ideal of R and M be a non-zero finitely generated R-module. In this paper we show that Huneke's two different conjectures are equivalent. Also we provide some partially affirmative answers to them. In fact it is shown that the Bass numbers of [Formula: see text] are finite for all i ≥ 0, whenever d ≤ 3. Also if (R, m) be regular local ring, we show that the Bass numbers [Formula: see text] are finite, for all i ≥ 0 and all j ≥ 0, and [Formula: see text], for all i ≥ 0 and all j ≥ 0, whenever height(I) = 1.

Some model theory of Bezout difference rings- a survey

This survey article is based on a joint work with Ehud Hrushovski on some Bezout difference rings ([19]). We study in a model-theoretic point of view certain classes of difference rings, namely rings with a distinguished endomorphism and more particularly rings of sequences over a field with a shift. First we will recall some results on difference fields. A difference field is a difference ring which is a field (in this case, the endomorphism is necessarily injective); and one can show that any difference field can be embedded in an inversive difference field, namely a field with an automorphism ([5]). The model theory of difference fields started in the nineties; one motivation was to understand the model-theory of non principal ultraproducts of the algebraic closure Fp of the prime fields endowed with the Frobenius maps. The existence of a model-companion for the theory of difference fields was shown by L. van den Dries and A. Macintyre ([27]). Here, we will recall the geometric axiomatization ACFA of the class of existentially closed difference fields given by Z. Chatzidakis and E. Hrushovski ([7]); they identified the different completions of ACFA, and deduce its decidability. The proofs that the non principal ultraproducts of the Fp endowed with the Frobenius maps, are models of ACFA, are much more difficult (?). An easy consequence of the decidability of ACFA is the decidability of some von Neumann commutative regular difference rings. Indeed, these are Boolean products of fields and whenever the endomorphism σ fixes the prime spectrum of the ring, we may apply transfer results due to S. Burris and H. Werner in these products ([4]). More generally we will consider Bezout difference rings in the point of view

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.

Nontriviality of SK1( R[ M

The main result of this paper is as follows: For any commutative regular (actually even K 2-regular) ring R and any finitely generated intermediate monoid Z + r ⊂ M ⊂ Q + r (for some natural r) the following conditions are equivalent: 1. (a) M ≈ Z + r 2. (b) R[ M] is K 1-regular 3. (c) M is seminormal and SK 1( R) = SK 1( R[ M]) (i.e. the natural homomorphism SK 1( R) → SK 1( R[ M]) is an isomorphism) and, if in addition Ω R ≠ 0 4. (d) SK 1( R) = SK 1( R[ M]) where ω R is a module of absolute differentials. The implications ( a) ⇒ ( b) ⇒ ( c) are well known. In Sections 8–10 we present examples, further generalizations and applications.

The nil radical of a fuzzy ideal

Abstract We define ab initio the fuzzy nil radical of an ideal of a commutative ring R . The definition has been successful in establishing the analogues of most of the fundamental ground results involving radicals in the fuzzy setting. The concept of a fuzzy semiprime ideal of a commutative ring R is introduced through the fuzzy nil radical of an ideal. A commutative regular ring is characterized by its semiprime ideals. At the end we demonstrate that the intersection of all fuzzy semiprime ideals of a commutative ring which contain a given fuzzy ideal λ is precisely the nil radical √λ of λ. This establishes the vital connection between a fuzzy semiprime ideal and the nil radical of a fuzzy ideal.

On Boolean algebras and integrally closed commutative regular rings

AbstractIn this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai, partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S′(ai, l) for l &lt; n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ϵ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.

Commutative regular rings and Boolean-valued fields

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring (the original proof is due to Carson [2]).Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. (Lemma 4.2 shows that their equality relation is the same as the one used in this paper.) To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space (see, for example, [9] and [8]). These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory.

Noetherian Rings of Finite Global Dimension

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

On free commutative regular rings

It has been observed that the category of all regular rings, when viewed as a full subcategory of the category of all rings, is not an algebraic variety. However if a regular ring is viewed as a ring equipped with a unary operation q such that xxqx = x and 0000q= 0, then the category of all rings with this added structure is indeed a variety but it is not, in any natural way, a subcategory of the category of all rings. In this article regular rings are viewed in this way and the free commutative regular rings are constructed. They are derived from the universal commutative regular ring associated with certain polynomial rings.

Coherence and weak global dimension of R[[X]] when R is von Neumann regular

dimension of R. Very little is known about what can occur except that the weak global dimension must increase by at least one and does increase by exactly one when R[[X]] is coherent. This paper attempts to cast some light on this problem and the closely related question of the stability of coherence under the formation of the power series ring. It is devoted to determining necessary and sufficient conditions on a commutative (von Neumann) regular ring R for the power series ring R[[X]] to b e coherent (equivalently, semihereditary) and also conditions for R[[X]] t o h ave weak global dimension one. Surprisingly, it turns out that R[[X]] h as weak global dimension one precisely when R[[X’j] is a B&out ring so that property is characterized as well. Each of these properties is characterized in several ways both in terms of internal conditions on R and conditions that involve the category of R-modules. Perhaps the conditions which can be most readily verified for a specific ring are expressed in terms of a natural partial order z< which is defined on R by a < b if and only if ab = a2 for a, b in R. For example, it is shown that R[[X]] is coherent if and only if every countable subset of R that forms a chain in the partial order < on R has a least upper bound in R. As further illustrations of our results we mention that R[[Xj] h as weak global dimension one precisely in case R satisfies either of two limited forms of self-injectivity and also in case R satisfies a restricted type of algebraic compactness. These results permit us to give an example that shows R[[X]] can have weak global dimension one without being coherent even when R is a Boolean ring. This settles a question raised by Jensen [7]. Another example is included to illustrate the added complexity that occurs in arbitrary regular rings as compared with Boolean rings. This example also provides a pair of regular rings R contained in S, and hence a faithfully flat extension, such that R[[X]] is not pure in S[[X]]. Thus it gives a negative

Integral Closures of Uncountable Commutative Regular Rings

Necessary and sufficient conditions are given for a commutative regular ring to have a prime integrally closed extension.

Integral closures of uncountable commutative regular rings

Necessary and sufficient conditions are given for a commutative regular ring to have a prime integrally closed extension.

Commutative Regular Rings with Integral Closure

First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient.