A decomposition theorem for ℘∗-semisimple rings

✍ Scribed by Hai Quang Dinh; Dinh Van Huynh

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 235 KB
• Volume: 186
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(03)00127-0

✦ Synopsis

A module M is said to satisfy the condition (˝ * ) if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisÿes (˝ * ) are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right ˝ * -semisimple rings. Right ˝ * -semisimple rings are right artinian. However, in general, a right ˝ * -semisimple rings need not be left ˝ * -semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left ˝ * -semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.