On Some Classes of Artinian Rings

✍ Scribed by Dinh Van Huynh; S.Tariq Rizvi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 152 KB
• Volume: 223
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8080

✦ Synopsis

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. A ring R is called CS-semisimple if every right R-module is CS. For a ring R, we show that:

Ž . 1 R is right artinian with Jacobson radical cube zero if every countably generated right R-module is a direct sum of a projective module and a CS-module.

Ž .

Ž . 2 The following conditions are equivalent: i Every countably generated right R-module is a direct sum of a projective module and a quasicontinuous Ž . module; and ii every right R-module is a direct sum of a projective module and a quasi-injective module.

Ž .

We describe the structure of rings in 2 and show that such a ring is not necessarily CS-semisimple. ᮊ 2000 Academic Press R w x 6, 12 for details on CS-modules. Ž Let ဧ be a property of modules over a ring R such as the property of being injective, being CS, or being a direct sum of a projective module