László Fuchs

Baer modules over domains

For a commutative domain R with 1, an R-module B is called a Baer module if ExtR (B, T) = 0 for all torsion R-modules T. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over hlocal Prufer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.

Baer modules over valuation domains

SummaryA module B over a commutative domain R is said to be a Baer module if ExtR1 (B, T)=0for all torsion R-modules T. The case in which R is an arbitrary valuation domain is investigated, and it is shown that in this case Baer modules are necessarily free. The method employed is totally different from Griffiths method for R=Z which breaks down for non-hereditary rings.

Uniserial modules over valuation rings

This is the first in a series of papers deahng with the theory of modules over valuation rings. It has been observed that a variety of concepts and basic results in abelian group theory can be extended, mutatis mutandis, to modules over valuation rings, and several other results, which require drastic modification for valuation rings, can also be dealt with by suitably generalizing ideas of abelian group theory. Our purpose is to develop techniques for modules over valuation rings; actually, these are the simplest kind of commutative non-noetherian rings. Our point of departure is abelian group theory in the local case when the groups are merely modules over Z,, the integers localized at a prime p, which is a discrete, rank one valuation domain. In the process of generalization, the most attractive and frequently used properties are sacrilied, pleasant properties we are so accustomed to in abelian groups are gone; in return, we learn new features and discover new phenomena in the behavior of modules. Several results are known on modules over valuation rings R; see the references [3-lo] or the survey article 121. This paper is devoted to the study of uniseriul R-modules, i.e., those R-modules in which the submodules form a chain. As far as the simplicity of the structure is concerned, these are second only to cyclic modules. They have been investigated by Shores and Lewis [9]; we make use of their results, especially, their description of endomorphism rings. We study various aspects of these modules with special emphasis on their quasiand pure-injectivity, as well as on the existence of pure uniserial submodules in torsion R-modules. In some cases, as expected, more explicit results can be established only under the additional hypothesis that R is almost maximal.

On divisible modules over domains

This note is devoted to the study of divisible modules over arbitrary commutative domains R with 1. Whereas divisible modules over Dedekind domains can be completely characterized by numerical invariants, not much is known about their structures in the general case. Divisible modules have been studied by Matlis [6] who was the first to distinguish between divisibility in general and h-divisibility. He established an important duality between h-divisible torsion modules and complete torsion-free R-modules [7] . Matlis [6] and Hamsher [4] characterized those domains R for which all divisible R-modules are h-divisible, e.g. by the property that the field Q of quotients of R has projective dimension 1.

Infinite Goldie dimensions

Let R be a ring with identity and M a unital right R-module. By the finite Goldie dimension of M is meant the largest integer m (provided it exists) such that M contains the direct sum of m nonzero submodules. If no such m exists, then M is said to have infinite Goldie dimension. Until recently, not much attention has been given to the case of infinite Goldie dimensions, though the definition is immediate. The Goldie dimension of M-denoted Gd M-is the supremum 1, of all cardinals K such that M contains the direct sum of K nonzero submodules. We believe that one of the main reasons why infinite Goldie dimensions have been ignored is that suprema are difficult to handle, and there is no guarantee that a module M of Goldie dimension 1. contains the direct sum of exactly 2 nonzero submodules. Given a cardinal number K, we say K is attained in M if M contains a direct sum of K nonzero submodules. If K is not a limit cardinal, i.e., if it is of the form tc=H,+, for some ordinal CL, then K 2 Gd M is attained in M. Recall that an infinite cardinal K is called regular if ~~ < K for iE Z with IZl < K implies c K~ < K. Otherwise it is called singular. An uncountable, regular, limit cardinal is said to be inaccessible ([HJ, p. 1631 or [L, p. 1373). The reader is reminded that the existence of inaccessible cardinals cannot be proved in ZFC (Zermelo-Fraenkel set theory with Axiom of Choice added), and that in the constructible universe, there are no such cardinals. All proofs here are within the framework of ZFC. Our purpose here is to show that if Gd M is not an inaccessible cardinal, then Gd M is attained in M. Our result is best possible; for inaccessible cardinals see the remark at the end.

Commutative ideal theory without finiteness conditions: Primal ideals

Our goal is to establish an efficient decomposition of an ideal A of a commutative ring R as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: A = ∩ P ∈ xA A (P) , where the A (P) are isolated components of A that are primal ideals having distinct and incomparable adjoint primes P. For this purpose we define the set Ass(A) of associated primes of the ideal A to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irredundant, or residually maximal, or the unique representation of A as an irredundant intersection of isolated components of A. Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for P ∈ Spec R that an ideal A C P is an intersection of P-primal ideals if and only if the elements of R\P are prime to A. We prove that the following conditions are equivalent: (i) the ring R is arithmetical, (ii) every primal ideal of R is irreducible, (iii) each proper ideal of R is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum. In particular, the integral domains having these equivalent properties are the Prufer domains possessing a certain property.

Butler groups of arbitrary cardinality

We show that in the constructible universe, the two usual definitions of Butler groups are equivalent for groups of arbitrarily large power. We also prove that Bext2(G, T) vanishes for every torsion-free groupG and torsion groupT. Furthermore, balanced subgroups of completely decomposable groups are Butler groups. These results have been known, under CH, only for groups of cardinalities ≤ ℵω.

A WEAKER FORM OF BAER'S SPLITTING PROBLEM OVER VALUATION DOMAINS

Abstract Eklof-Fuchs [3] have shown that over an arbitrary valuation domain R, the modules B which satisfy Ext 1/R (B,T) = 0 for all torsion R-modules T are precisely the free R-modules. Here we modify the problem and describe all R-modules B for which Ext 1/R (B, T) vanishes for all bounded and for all divisible torsion R-modules T. It is well known that if R is a descrete rank one valuation domain then all torsion—free R-modules B have this property.

On a class of butler groups

A complete classification, both up to quasi isomorphism and up to isomorphism, is given of the strongly indecomposable torsionfree abelian groups that occur as quotients of a finite rank completely decomposable torsionfree group modulo a rank 1 subgroup.

THE FULLY INVARIANT EXTENDING PROPERTY FOR ABELIAN GROUPS

An abelian group has the FI-extending property if every fully invariant subgroup is essential in a direct summand. A mixed abelian group has the FI-extending property if and only if it is a direct sum of a torsion and a torsion-free abelian group, both with the FI-extending property. A full characterization is obtained for the abelian groups with the FI-extending property which are either torsion-free of finite rank or torsion.