On simple-injective modules

For a right module [Formula: see text], we prove that [Formula: see text] is simple-injective if and only if [Formula: see text] is min-[Formula: see text]-injective for every cyclic right module [Formula: see text]. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is [Formula: see text] if and only if simple-injective right modules are projective. For a commutative Noetherian ring [Formula: see text], we prove that every finitely generated simple-injective [Formula: see text]-module is projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.