Abelian groups in which every endomorphism is a left multiplication

Abstract

Let <G+> be an abelian group. With each multiplication on G (binary operation * such that <G + *> is a ring) and each g e G is associated the endomorphism gf of left multiplication by g. Let L(G) = {gt\geG, * Mult G}. Abelian groups G such that L(G) = E{G) are studied. Such groups G are characterized if G is torsion, reduced algebraically compact, completely decomposable, or almost completely decomposable of rank two. A partial results is obtained for mixed groups. Proof. If L(G) = E(G), then there exists u e G, e Mult G such that