On hereditarily and super R-Hopfian and L-co-Hopfian abelian groups

An Abelian group is said to be R-Hopfian L-co-Hopfian] if every surjective injective] endomorphism has a right left] inverse. An Abelian group G is said to be hereditarily R-Hopfian hereditarily L-co-Hopfian] if each subgroup of G is R-Hopfian L-co-Hopfian]; similarly G is super R-Hopfian super L-co-Hopfian] if each homomorphic image of G is R-Hopfian L-co-Hopfian]. The various classes of hereditarily and super R-Hopfian and L-co-Hopfian groups are studied and necessary conditions for groups to have these properties are derived; in several, but not all, cases, su?cient conditions are also obtained.