Commutator invariant subgroups of Abelian groups.
The commutator \([\varphi,\psi]\) of two elements of a ring is the element \(\varphi\psi-\psi\varphi\). A subgroup \(H\) of an Abelian group \(A\) is commutator invariant if \([\varphi,\psi]H\subseteq H\) for all commutators in the endomorphism ring of \(A\). The purpose of this paper is to describe the commutator invariant subgroups of an Abelian group. In particular, the author shows that the commutator invariant subgroups of separable and algebraically compact torsion-free Abelian groups are fully invariant. The \(E\)-center of \(A\) is the subgroup annihilated by all commutators, and the \(E\)-commutant is the subgroup generated by the images of all commutators. The author describes the \(E\)-centers and the \(E\)-commutants of several notable classes of Abelian groups.
scientific article; zbMATH DE number 5824094
scientific article