Paul C. Eklof

Ultraproducts for Algebraists

Publisher Summary The ultraproduct construction is an algebraic operation whose importance derives from its model-theoretic properties. The algebraic character of the construction makes it attractive tool to employ in giving an account of applications of model theory to algebra. This chapter presents basic properties of ultraproducts and some of their applications to algebra. The chapter is divided into three parts. The first part, Basics, presents the definitions of ultrafilters and ultraproducts and proves the fundamental theorem of ultraproducts. The last part discusses the functorial properties of ultraproducts. The rest of the results are given in two parts, Compactness and Saturation, whose titles refer to the key model-theoretic properties of ultraproducts used in proving these results.

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.

On Whitehead modules

Abstract It is proved that it is consistent with ZFC + GCH that, for any reasonable ring R , for every R -module K there is a non-projective module M such that Ext R 1 ( M , K ) = 0; in particular, there are Whitehead R -modules which are not projective. This is generalized to show that it is consistent that, for certain rings R , there are Whitehead R -modules which are not the union of a continuous chain of submodules so that all quotients are small Whitehead R -modules. An application to Baer modules is also given: it is proved undecidable in ZFC + GCH whether there is a single test module for being a Baer module.

Tilting Cotorsion Pairs

Let R be a ring and T be a 1-tilting right R-module. Then T is of countable type. Moreover, T is of finite type in case R is a Prüfer domain.

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.

Dually slender modules and steady rings.

A module M over a ring R is called dually slender if Hom (M, —) commutes with direct sums of -modules. For example, any finitely generated module is dually slender. A ring R is called right steady if each dually slender right -module is finitely generated. We provide a model theoretic necessary and sufficient condition for a countable ring to be right steady. Also, we prove that any right semiartinian ring of countable Loewy length is right steady. For each uncountable ordinal σ, we construct examples of commutative semiartinian rings Γσ, and Qa, of Loewy length σ +1 such that Τσ is, but Q0 is not, steady. Finally, we study relations among dually slender, reducing, and almost free modules. 1991 Mathematics Subject Classification: 16D40, 16D90, 16E50, 03C60.

Categoricity results for L∞κ

Abstract Let V denote a variety of algebras in a countable language. An algebra is said to be L∞κ-free if it is L∞κ-equivalent to a (V-) free algebra. If every L∞ω1-free algebra of cardinality ω1 is free, then for all infinite cardinals κ every L∞κ-free algebra of cardinality κ is free. Further, assuming suitable set-theoretic hypotheses, if there is a non-free L∞ω1-free algebra of cardinality ω1, then for a proper class of cardinals κ there are non-free L∞κ-free algebras of cardinality κ. The varieties in which the class of free algebras are definable by a sentence in Lω1ω are characterized as the weak Schreier varieties in which every L∞ω-free algebra of cardinality ω1 is free. A weak Schreier variety is one in which every L∞ω-elementary substructure of a free algebra is free. In fact, assuming suitable set-theortic hypotheses, for weak Schreier varieties the class of free algebras is definable in L∞∞ iff it is definable in Lω1ω. Varieties in uncountable languages are also considered.

The Kaplansky test problems for ℵ₁-separable groups

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for א1-separable abelian groups of cardinality א1. In fact, there is an א1-separable abelian group M such that M is isomorphic to M⊕M⊕M but not to M⊕M . We also derive some relevant information about the endomorphism ring of M .

On singular compactness

We present some applications of Shelahs singular compactness theorem to algebraic situations where the Shreier property fails. The principal application is to valuated vector spaces, where we make use of an alternate, unpublished, version of Shelahs theorem.

A non-reflexive Whitehead group ☆

Abstract We prove that it is consistent that there is a non-reflexive Whitehead group, in fact one whose dual group is free. We also prove that it is consistent that there is a group A such that Ext (A, Z ) is torsion and Hom (A, Z )=0 . As an application we show the consistency of the existence of new co-Moore spaces.