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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, suﬃcient conditions are also obtained.
Brendan Goldsmith & Ketao Gong (2018) On hereditarily and super R-Hopfian and L-co-Hopfian abelian groups, Communications in Algebra, 46:5, 1889-1901, DOI: 10.1080/00927872.2017.1360332