Weakly injective modules and weakly projective modules

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

Whitehead modules

We define the class of almost <TEX>${\omega}_1-p^{\omega+n}$</TEX>-projective abelian p-primary groups and investigate their basic properties. The established results extend classical achievements due to Hill (Comment. Math. Univ. Carol., 1995), Hill-Ullery (Czech. Math. J., 1996) and Keef (J. Alg. Numb. Th. Acad., 2010).

The computation of odd dimensional projective surgery groups of finite groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

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

Projective modules and class groups

Let p be a prime, R a commutative ring of characteristic p , G a p -primary abelian group, and V(RG) the group of normalized units in the group algebra RG . It is largely determined when V(RG) is totally projective. This generalizes the well-known special case when R is a perfect field. Moreover, the current proof has the advantage of greater clarity.

We provide descriptions for the moduli spaces Rep(Γ, PU(m)), where Γ is any finitely generated Abelian group and PU(m) is the group of m × m projective unitary matrices. As an application, we show that for any connected CW-complex X with π1(X)≅Zn, the natural map π0(Rep(π1(X), PU(m))) → [X, BPU(m)] is injective, hence providing a complete enumeration of the isomorphism classes of flat principal PU(m)-bundles over X.

Skew pure projective abelian groups

Quasi-pure projective and injective torsion free abelian groups of rank 2

In this paper R R denotes an associative ring with an identity, and all modules are unital left R R -modules. It is shown that the existence of a quasi-projective cover for each module implies that each module has a projective cover. By a similar technique the following statements are shown to be equivalent: 1. R R is semisimple and Artinian; 2. Every finitely generated module is quasi-projective; and 3. The direct sum of every pair of quasi-projective modules is quasi-projective. Direct sums of quasi-injective modules are also investigated.

Let G-ZFIDCW denote the category of G-spaces having the G-homotopy type of a finitely dominated G-CW complex for a compact Lie group G. Luck [8] has introduced a functor Wa from G-EF^C^ into the category of abelian groups and has realized the equivariant finiteness obstriction as the element w(X) in Wa(X). That is, a finitely dominated G-CW complex X is G-homotopy equivalent to a finite G-CW complex if and only if w(X)=Q. When G is the trivial group, there is an isomorphism from Wa(X) to the reduced projective group KQ(Z[πι(X)]) which sends the element w (X) to the WalΓs finiteness obstruction ([14]). Anderson [1] and Ehrlich [4] have studied a sufficient condition for w(E)= 0 for some fibration E-^B with fiber S. Munkholm, Pedresen [11], Luck [6, 7, 9] and others have studied the transfer map KQ(Z[^(B)])-^K:Q(Z[^(E)]). The purpose of this paper is to get a sufficient condition for w(X)=0 for a S^space X and a finite cyclic group L. We call G-maps /0: YQ-^X and /0: Y4-*X equivalent if there exists a commutative diagram