H. Pat Goeters

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.

FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor1 (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.

Warfield Duality and Module Extensions Over a Noetherian Domain

The relationship between the torsion-free property of certain extension functors and a duality defined by Warfield is explored. We show that Warfield duality holds over a Noetherian integral domain whose integral closure inside the quotient field is finite over the ring, if and only if every ideal can be generated by two elements.

A dual to Baer’s lemma

Let A be an abelian group. We investigate the splitting of sequences (*) 0 -+ P -+ G -+ H -+ 0 with P A-projective: Examples show that restrictions on G and H must be imposed to obtain a dual to Baers Lemma. A characterization of the splitting of sequences like (*) where G is A-reflexive and RA(H) = 0 is given in terms of A and E(A), when A is slender and nonmeasurable. Furthermore, we consider related problems and present applications of our results.

The Structure of Ext for Torsion-Free Modules of Finite Rank over Dedekind Domains

Abstract The structure of Ext is fundamental to understanding rings and their modules. For example,relative injectivity can be interpreted as a structural property of Ext as in when (A,B) is torsion-free or zero. The purpose of this work is to examine (A,B) under the assumption that A and B are torsion-free modules of finite rank,and R is a Dedekind domain. It is the abelian group theorists mantra that their results extend to modules over Dedekind domains. This is almost always the case,however the structure of Ext is strongly influenced by the rank of the completion of R; unlike the case for the integers,the completion of a domain may have finite rank over the domain. Below,R will represent an integral domain with quotient field Q,and we will assume that R ≠ Q. In most case R will be assume to be Dedekind. All unadorned Ext,Hom,and ⊗ symbols are with respect to R.

Reflexive rings and subrings of a product of Dedekind domains

Abstract An extension of Warfield duality for modules over rings whose integral closure is a finite product of Dedekind domains is given. Our main example illustrating the concepts also substantiates a conjecture of Matlis.

Duality and self-reflexive torsion-free abelian groups

We characterize the torsion-free abelian groups G of finite rank such that Hom(−, G) defines a rank preserving duality on the category of End(G)-submodules of finite sums of copies of G. Our results provide a maximal extension of Warfields duality result for torsion-free groups of finite rank. In order to obtain our extension, we study the torsion-free abelian groups G for which G ≅nat Hom(Hom(G, G), G).