Torsion theoretic points and spaces

A module M with Krull dimension is said to satisfy the large condition if for any essential submodule L of M, the Krull dimension of M/ L is strictly less than the Krull dimension of M. For a right noetherian ring R with Krull dimension a this is equivalent to the condition that every f.g.uniform submodule of E(RR) with Krull dimension a is critical.It is also shown that if R is right noetherian with Krull dimension a and if I0 is a right ideal maximal with respect to K dim 10 < a, then R satisfies the large condition if and only if I0 is a finite intersection of cocritical right ideals and I0 is closed.

We show that a simple Noetherian ring of finite global dimension and Krull dimension one is Monta equivalent to a domain.In [4] is given an example of a simple Noetherian ring that is not Morita equivalent to a domain.Now the ring in question has infinite global dimension but Krull dimension one.In contrast, we show here that if R is a simple Noetherian ring of finite global dimension and Krull dimension one, then R is Morita equivalent to a domain.This is a corollary of Theorem 1, which gives a generalisation of this result to simple rings of arbitrary finite Krull dimension.Throughout this note all rings will contain an identity and all modules will be unitary.We will denote the uniform dimension of a module M by ud(M) and the projective dimension by pd(M).The left Krull dimension of a ring R

In this paper, we discuss sections of higher homotopy groupoid bundles those yield by group actions. We study some basic properties related to the left-invariant (G-invariant) of such sections and also present some conditions for such sections to be left-invariant (G-invariant). We concentrate a kind of group action on the Core higher homotopy groupoid bundle of certain topological spaces which undergo a regular topological group action. Interestingly, each element of orbit space yields continuous sections of the Core higher homotopy groupoid bundle having beautiful algebraic properties. Moreover, each of those continuous sections behaves nicely as a left-invariant section with respect to topological group action by G. Since higher homotopy groups are abelian, the set of such sections forms always an abelian group. Moreover, each element of the group is a left-invariant section, therefore, it is convenient to call them commutative group of left-invariant sections of the Core higher homotopy groupoid bundle. Further, we show this group is a topological invariant and study some basic properties of it. At the end of the paper, we give interrelations between the retraction map and this group.

The (right) Ziegler spectrum of a ring R is a topological space the points of which are the isomorphism types of the indecomposable pure-injective (right) K-modules and is denoted by Z g R. The CB-rank of the Ziegler spectrum of a ring R is a measure for the complexity of the cat­egory of K-modules. The m-dimension of a module is a measure for the complexity of the lattice of its pp-definable subgroups. Using the clas­sification of the isomorphism types of the indecomposable pure-injective modules over a serial ring obtained by Eklof, Herzog and Puninski, we show that if R is a serial ring with Krull dimension, then Z g R has CB-rank and for every point in ZgR, its CB-rank is equal to its m-dimension. We also show that a serial Krull-Schmidt ring R has Krull dimension iff the largest theory of (right) R-modules has m-dimension.

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring [Formula: see text] First, we study the prime spectrum of R. We prove that the contraction map: Spec (A[[X]]) → Spec (R); Q ↦ Q ∩ R induces a homeomorphism, for the Zariski's topologies, from {Q ∈ Spec (A[[X]]) | XI[[X]] ⊈ Q} onto {P ∈ Spec (R) | XI[[X]] ⊈ P}. If P ∈ Spec (R) is such that XI[[X]] ⊆ P then there exists p ∈ Spec (A) such that P = p + XI[[X]]. Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim (R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim (R) = dim (A[[X]]) = dim (A) + 1. Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

Totally acyclic complexes

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Năstăsescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal.

We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules. Though originally defined by Gulliksen[7] in the 1970s, this extension has seldom been used in subsequent research. Despite this neglect, we will show that the ordinal valued length is a quite natural measure of the size of a Noetherian module, and has advantages over more familiar measures such as uniform dimension, Krull dimension, and reduced rank. We will also demonstrate how some familiar properties of left Noetherian rings can be proved efficiently using length and the arithmetic properties of ordinal numbers.

ABSTRACTWe extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.

This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle.

For a commutative ring A we study here the Grothendieck group, K0A, of finitely generated projective A-modules, and the Picard group, Pic (A), of projective modules of rank one (under (?A). Writing koA for the elements of rank zero in KOA, there is an epimorphism det: K0A Pic (A) defined by the rth exterior power of a projective module of rank r. Our original objective was to calculate KJ(Zw), where w is a finitely generated abelian group. We do this in ? 8. The bulk of the paper is a rather general discussion of the techniques, some already familiar in other contexts, which we use to make this calculation. Our procedure is to decompose w as w0 x T, with w0 a finite group and T free abelian, and then to write Zw= A[T] where A = Zw0. Then A is a noetherian ring of (Krull) dimension one. For such an A we show (Theorem 7.8) that det: ko(A[T]) Pic (A[T]) is an isomorphism. In ?9 we prove also a non-stable analogue of this for T of rank <2. In ?8 we show (Theorem 8.1) that

Partial cross sections for electron capture in collisions of Ar6+ ions with atomic deuterium at an impact energy of 3.27 keV have been calculated in a 26-state atomic-basis expansion. The results are at variance with the final-state distribution deduced from the experimental energy-gain spectrum by a standard procedure. To perform a more direct comparison with the experimental data, the authors have constructed differential capture cross sections from the calculated probability amplitudes in the eikonal approximation and, in turn, a theoretical energy-gain spectrum. At this level there is good agreement between theory and experiment. It is demonstrated that capture to the more strongly bound final states in the energy-gain spectrum appears as overlapping broad and skew structures, not compatible with the simple lineshape presumed in the standard data reduction.

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T π-compact provided G/T πG is π-complete, extending the abelian group result in a natural way.