Commutative rings for which every continuous module is quasi-injective
A module is called continuous if (i) every submodule is essential in a summand and \((ii)\quad every\) submodule, isomorphic to a summand, is itself a summand. It is well known that every quasi-injective module is continuous. The purpose of this paper is to study commutative rings for which every continuous module is quasi-injective. It is shown that large classes of commutative rings, such as noetherian rings, and semiprimary rings with Jacobson radical square zero, have this property. We provide necessary and sufficient conditions, for some classes of commutative rings such as finitary rings, perfect rings, and chain rings to have this property. A necessary and sufficient condition for an arbitrary commutative ring, for which every uniform continuous R-module is quasi- injective, is also given.
scientific article; zbMATH DE number 4025563