A study on additive endomorphisms of rings (Q2712548)

Let \(R\) be an associative ring, not necessarily with identity, let \(\text{End }R\) denote the set of all ring endomorphisms of \(R\), and let \(\text{End}(R,+)\) be the set of all endomorphisms of the additive group of \(R\). The author is motivated by Sullivan's Problem which asks for a characterization of \(AE\)-rings, i.e. those rings \(R\) for which \(\text{End }R=\text{End}(R,+)\). A ring \(R\) is termed an \(AEG\)-ring if \(\text{End}(R,+)\) is generated, as an additive group, by the set \(\text{End }R\). If \(R=\bigoplus_{i=1}^n A_i\) is the ring direct sum of finitely many subrings \(A_i\), it is shown that \(R\) is an \(AEG\)-ring iff, for each \(i,j\), every \(f\in\Hom_\mathbb{Z}(A_i,A_j)\) is a \(\mathbb{Z}\)-linear combination of ring homomorphisms from \(A_i\) to \(A_j\). Related topics are explored, e.g. the connection with \(LSD\)-generated rings, and several examples are provided.