Does the automorphism group generate the endomorphism ring in \(\text{Rep}(S,R)\)?
In 1963 Laszlo Fuchs posed the question of when the automorphism group of an Abelian group additively generates the endomorphism ring. Since then Fuchs's problem and related generalizations have been extensively studied in categories such as Abelian groups, vector spaces, free modules over PID's, general rings, and Warfield modules. The current work investigates this topic in the setting of representations of a commutative ring \(R\) over a finite partially ordered set \((S,\leq)\). The question is answered fully when \(R\) is a field (\(\not=\text{GF}(2)\)) and partial results are obtained when \(R\) is a principal ideal domain which is not a field. An element \(\underline V\) of the category \(\text{Rep}(S,R)\) is a family \(\underline V=(V,V_i\mid i\in S)\) where \(V\) is a free \(R\)-module and each \(V_i\) is a free \(R\)-submodule of \(V\) such that \(V_i\subseteq V_j\) whenever \(i\leq j\). A homomorphism \(\underline\phi\) from \(\underline V\) to \(\underline W\) is a homomorphism \(\phi\colon V\to W\) such that \(\phi|_{V_i}\) maps \(V_i\) to \(W_i\). One of Strüngmann's foremost results states that when \(R=k\) is a field (\(\not=\text{GF}(2)\)) and either \(V\) is a finite dimensional vector space over \(k\) or \(|S|\leq 3\), then \(\text{End}_k(\underline V)=\Aut_k(\underline V)+\Aut_k(\underline V)\). When \(|S|>3\), the author demonstrates that anything can happen with regard to Fuchs's property.
scientific article; zbMATH DE number 1525270