Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

Noetherian Skew Inverse Power Series Rings

We study skew inverse power series extensions R[[y ^− 1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ-derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension are all equal to 2n.     

On s-Semipermutable Maximal and Minimal Subgroups of Sylow p-Subgroups of Finite Groups

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R?_k k(X) in terms of the heights of simple modules with large endomorphism rings.     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

Characterizations of some rings with <Emphasis Type="Italic">C</Emphasis>-projective, <Emphasis Type="Italic">C</Emphasis>-(FP)-injective and <Emphasis Type="Italic">C</Emphasis>-flat modules

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.     

THE REGULARITY DEGREE AND EPIMORPHISMS IN THE CATEGORY OF COMMUTATIVE RINGS

Elements of the universal (von Neumann) regular ring T(R) of a commutative semiprime ring R can be expressed as a sum of products of elements of R and quasi-inverses of elements of R. The maximum number of terms required is called the regularity degree, an invariant for R measuring how R sits in T(R). It is bounded below by 1 plus the Krull dimension of R. For rings with finitely many primes and integral extensions of noetherian rings of dimension 1, this number is precisely the regularity degree.

For each n ≥ 1, one can find a ring of regularity degree n + 1. This shows that an infinite product of epimorphisms in the category of commutative rings need not be an epimorphism.

Finite upper bounds for the regularity degree are found for noetherian rings R of finite dimension using the Wiegand dimension theory for Patch R. These bounds apply to integral extensions of such rings as well.     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.     

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.     

Some Results on Injective Modules Over a Ring of Krull Type

ABSTRACT

In this article, we study injective modules over a ring of Krull type A. Our main result is E(K/A)? ?_{ω∈Ω ^t} E(K/?_ω), where Ω ^t is a thin defining family of valuations of A. We also characterize the rings of Krull type A such that TE(K/A) is a cogenerator of the quotient category Mod(A)/?₀, where ?₀ is the thick subcategory of the modules with trivial maps into the codivisorial modules.     

Etude de la categorie des algebres de Hopf commutatives connexes sur un corps

Let k be a perfect field of characteristic p0; the categoryH of connected abelian Hopf algebras over k is abelian and locally noetherian. Technics of locally noetherian categories are used here to obtain Krull and homological dimensions ofH (which are respectively 1 and 2), and a decomposition ofH in a product of categories. First we have, whereH ^– is the category of Grassman algebras, andH ⁺ consists of Hopf algebras which are zero in odd degrees; then we prove thatH ⁺ itself is a product of isomorphic categoriesH _n, n^*, and we give an equivalence betweenH _n and a category of modules. This is compared to some results of algebraic geometry about Greenberg modules.     

On ℵ0-Injectivity

Abstract

In this note we investigate ?₀-injectivity of rings and modules and review the literature around this topic. We observe that several characterizations of rings by injectivity can be expressed by ?₀-injectivity. Moreover we point out that the following three classes of rings are not axiomatizable: the right ?₀-self-injective rings, the right ?₀-self-injective regular rings and the regular Baer rings.     

Construction of Modules with Finite Homological Dimensions

A new homological dimension, called G^*-dimension, is defined for every finitely generated module M over a local noetherian ring R. It is modeled on the CI-dimension of Avramov, Gasharov, and Peeva and has parallel properties. In particular, a ring R is Gorenstein if and only if every finitely generated R-module has finite G^*-dimension. The G^*-dimension lies between the CI-dimension and the G-dimension of Auslander and Bridger. This relation belongs to a longer sequence of inequalities, where a strict inequality in any place implies equalities to its right and left. Over general local rings, we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence.     

Almost finite modules

Assume that ?(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local rings R and S. We say that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ?). We investigate the homological behaviour of such modules, as well as various properties of the rings R and S in the presence of an almost finite module of finite flat dimension over R.     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

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.     

Uniform rank over schmidt differential operator rings

This paper is concerned with the question of when a Schmidt differential operator ring S over a ring R must have the same uniform rank or reduced rank as R. Also, some information about those prime ideals of R which are invariant under a Schmidt higher derivation is derived. All rings in this paper are associative with unit and all modules are unital right modules.

In [1], Bell and Goodearl proved that for a Poincaré-Birkhoff-Witt extension T of a ring R, the rank of T and the rank of R agree when R is a right noetherian ring with no Z-torsion which is tame as a right module over itself. In this paper, we show that for a Schmidt differential operator ring S over a right noetherian ring R with no Z-torsion which is tame as a right module over itself the rank of S and the rank of R agree. Also, for any right noetherian R, it is proved that R_R and S_S have the same reduced rank.     

Modules semicocritical with respect to a torsion theory and their applications

The concepts of primitive ideal and semicocritical module with respect to a torsion theory are studied and related to the structure of torsionfree injective modules. Applications are made to the study of (1) composition series with respect to a torsion theory and (2) the structure of endomorphism rings of torsionfree modules. These results are natural generalizations of the properties of certain modules over (noetherian) rings with Krull dimension.     

Projective modules over the real algebraic sphere of dimension 3

Let A be a commutative noetherian ring of Krull dimension 3. We give a necessary and sufficient condition for A-projective modules of rank 2 to be free. Using this, we show that all the finitely generated projective modules over the algebraic real 3-sphere are free.