Additive groups of trivial near-rings
A near-ring on a group \(G\) is trivial if there exists a subset \(S \subseteq G\) such that multiplication in \(G\) is defined by \(x.y = y\) if \(x \in S\), and is 0 otherwise. A group \(G\) is a TNR (trivial near-ring)-group if every near-ring on \(G\) is trivial. If \(|G|\leq 2\) then \(G\) is a TNR-group. Are there any others? The main theorem states that if \(G\) has an endomorphism, not the identity or zero which is either idempotent, or has square 0 or the identity, then \(G\) is not a TNR-group. Among other consequences the author shows that the only finite or abelian TNR-groups are those of order 1 or 2.
scientific article; zbMATH DE number 878090