On Simple Noetherian Rings

• 来源期刊: 代数集刊（英文版）
• 关键词: simple rings prime rings noetherian rings artinian rings
• 年，卷（期）: 2003,（2）
• 页码范围: 135-140
• 页数: 6页
• 语种: 英文

摘要

A module M is called a CS-module (or extending module [5]) if every submodule of M is essential in a direct summand of M. It is shown that (i) a simple ring R is right noetherian if and only if every cyclic singular right R-module is either a CS-module or a noetherian module; (ii) for a prime ring R, if every proper cyclic right R-module is a direct sum of a quasi-injective module and a finitely cogenerated module, then R is either semisimple artinian or a right Ore domain; and (iii) a prime ring R is right noetherian if and only if every cyclic right R-module is a direct sum of a quasi-injective module and a noetherian module.