Elias Toubassi

Classifying Endomorphism Rings of Rank One Mixed Groups

We are going to prove two theorems on endomorphism rings of rank one abelian groups with simply presented torsion. In Theorem 1, we describe when the endomorphism rings of two groups are isomorphic, and in Theorem 2, we tell when the endomorphism ring of a group possesses only inner automorphisms. In the proofs we utilize two results: the first reduces the global problem to endomorphism rings of local groups; the second, a local theorem, classifies isomorphisms of endomorphism rings of local groups. The bulk of the proofs are then devoted to relating such isomorphisms to p-indicators.

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

Abstract. Warfield modules and simply presented modules are considered whose torsion submodule is a direct sum of cyclics. A correspondence between Warfield modules and completely decomposable groups is established by showing that such a mixed module G is Warfield if and only if

Endomorphisms of abelian groups and the theorem of Baer and Kaplansky

G/p^{\omega }G

Endomorphisms of rank one mixed modules over discrete valuation rings.

is simply presented and

A result on problem 87 of L. Fuchs

p^{\omega }G

A Splitting Criterion for a Class of Mixed Modules

is completely decomposable. We give an example to show that this result cannot be extended to ordinals

Isomorphisms of endomorphism rings of rank one mixed groups

\sigma >\omega

The module structure of

. We also connect with the work of Nunke to characterize modules G whose zeroth Ulm factors are torsion in terms of

{\rm Ext}\ (F,\,T)

G/p^\sigma G

over the endomorphism ring of

simply presented and