# E-rings, the functor Ext and a problem by Fuchs

Project funded by the German Research Foundation

The idea of the project is a collaboration with Professor Dr. Phil Schultz from the University of the Western Cape (Australia). The project focuses on three topics from abelian group theory:

We will investigate properties of groups that are induced by the action of homological functors like Hom(-,-) and Ext(-,-) on them. Firstly, (non-)commutative E-rings, which play an important role in homotopy theory since they are exactly the localizations of the ring of integers, will be studied. Recall that a ring is an E-ring if it is naturally isomorphic to the endomorphism ring of its own additive group. We aim at giving a new construction method for non-commutative E-rings as well as structure theorems for commutative E-rings.

A second topic will be properties of the functor Ext in the category of abelian groups. A main focus will be on the question when Ext commutes with direct sums and on the structure of Ext(G,H) in certain models of ZFC. This is related to Shelah's solution of the Whitehead problem.

The third topic of the project is the decomposition theory of abelian groups. Here, we will investigate mainly what direct decompositions can be realized for groups. One end is formed by the super decomposable groups, the other end is formed by the indecomposable groups. But what happens in between? Can we prescribe the ranks of the direct summands?

The project is funded by the German Research Foundation (DFG) for 6 six months and will start in November 2017.