Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

Subgroups of prescribed finite index in linear groups

LetR be a commutative ring,M a finitely generatedR-module andG a subgroup of Aut _R M. Under either of the following conditions, for every positive integerd there is a normal subgroupH ofG of finite index such thatG/H contains an element of orderd. (a)G is infinite and finitely generated. (b)R is finitely generated as a ring andG is not unipotent-by-finite. This extends recent work of A. Lubotzky.     

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each i 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)E \neq E(k), then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n^th cosyzygy has its Tor-depth equal to the depth of the ring whenever n 3 dn \geq d.     

On rings in which every maximal one-sided ideal contains a maximal ideal

Given a ring R, consider the condition: (^*) every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e ² = e ∈ R such that ele ? eRe every proper ideal I of R R satisfies (^*) if and only if eRe satisfies (^*). Hence with the help of some other results, (^*) is a Morita invariant property. For a simple ring R R[x] satisfies (^*) if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (^*), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring R R satisfies (^*) if and only if S satisfies (^*).     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

Local bijective gabriel correspondence and relative fully boundedness in τ-noetherian rings

Let Rbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of right R-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image of Rcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable right R-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime ideal Pthe annhilator of a finitely generated P-tame right R-module cannot be a prime ideal properly contained in P. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.     

Induced characters, Mackey analysis and primitive ideal spaces of nilpotent discrete groups

Let G be a nilpotent discrete group and Prim(C^*(G)) the primitive ideal space of the group C^*-algebra C^*(G). If G is either finitely generated or has absolutely idempotent characters, we are able to describe the hull-kernel topology on Prim(C^*(G)) in terms of a topology on a parametrizing space of subgroup-character pairs. For that purpose, we introduce and study induced traces and develop a Mackey machine for characters. We heavily exploit the fact that the groups under consideration have the property that every faithful character vanishes outside the finite conjugacy class subgroup.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

Module valuations and representation theory of orders I: Algebraic aspects

ABSTRACT

In this article, we study finitely generated reflexive modules over coherent GCD-domains and finitely generated projective modules over polynomial rings. In particular, we give a sufficient condition for a finitely generated reflexive module over a coherent GCD-domain to be a free module. By use of this result, we prove that every finitely generated projective R + [X]-module can be extended from R if R is a commutative ring with gl.dim(R) ≤ 2.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

On Finitely Annihilated Modules and Artinian Rings

Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only if every injective right R-module is finitely annihilated. This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the EPSRC for their financial support     

FP-RINGS

A ring R is called right FP-injective if every R-homomor-phism from a finitely generated submodule of a free right R-module F into R extends to F. In this paper a ring R will be called a right FP-ring if R is semiperfect, right FP-injective and has an essential right socle. The class of FP-rings strictly contains the class of right (and left) pseudo-Frobenius rings, and we show that it is right-left symmetric and Morita-invariant. As an application we show that if R is a left perfect right FP-injective ring, then R is quasi-Frobenius if and only if the second right socle of R is finitely generated as a right ideal of R. This extends the known results in the right selfinjective case.     

Some new dimensions of modules and rings

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Mat_m(R) is a right Köthe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.     

Homological dimensions and semidualizing complexes

For finite modules over a local ring and complexes with finitely generated homology, we consider several homological invariants sharing some basic properties with projective dimension. In the second section, we introduce the notion of a semidualizing complex, which is a generalization of both a dualizing complex and a suitable module. Our goal is to establish some common properties of such complexes and the homological dimension with respect to them. Basic properties are investigated in Sec. 2.1. In Sec. 2.2, we study the structure of the set of semidualizing complexes over a local ring, which is closely related to the conjecture of Avramov-Foxby on the transitivity of the G-dimension. In particular, we prove that, for a pair of semidualizing complexes X ₁ and X ₂ such that G _X2, we have X ₂ ≃ X ₁ ⊗ _R ^L RHom _R (X ₁, X ₂). Specializing to the case of semidualizing modules over Artinian rings, we obtain a number of quantitative results for the rings possessing a configuration of semidualizing modules of special form. For the rings with m ³=0, this condition reduces to the existence of a nontrivial semidualizing module, and we prove a number of structural results in this case. In the third section, we consider the class of modules that contains the modules of finite CI-dimension and enjoys some nice additional properties, in particular, good behavior in short exact sequences. In the fourth section, we introduce a new homological invariant, CM-dimension, which provides a characterization for Cohen-Macaulay rings in precisely the same way as projective dimension does for regular rings, CI-dimension for locally complete intersections, and G-dimension for Gorenstein rings. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 30, Algebra, 2005.     

Finitely presented and coherent ordered modules and rings

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features:.

(i) Every subring of Q, and every totally ordered division ring, is po-coherent.

(ii) For a partially ordered right module Aover a po-coherent poring R Ais po-coherent if and only if Ais a finitely presented .R-module and A ⁺is a finitely generated R ⁺-semimodule.

(iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent.

(iv) Every finitely po-presented abelian lattice-ordered group is po-coherent.     

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.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G _T=<1>,where G _T is the inertial group of R.