When Is a Simple Ring Noetherian? Original Research Article

A moduleMis known to be a CS-module (or an extending module) if every complement submodule ofMis a direct summand. It is shown that (i) a simple ringRmust be right noetherian if every cyclic singular rightR-module is CS, and (ii) over a simple ringRif every proper cyclic right module is quasi-injective, thenRis right hereditary and right noetherian.