Auslander–Reiten Theory for Artinian PI-Rings

✍ Scribed by Markus Schmidmeier

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 158 KB
• Volume: 207
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7448

✦ Synopsis

Let R be an artinian polynomial identity ring, that is an artinian ring whose radical factor R is an artin algebra. In this article, we investigate how the basic tools in the representation theory of artin algebras, in particular, dual and transpose, source and sink maps, and AR-quivers, can be applied to the study of the category of R-modules. It turns out that finitely generated indecomposable R-modules either behave like modules over an artin algebra with respect to several conditions from module theory and representation theory or do not satisfy any of these conditions at all. THEOREM 1. Let R be an artinian PI-ring and M a finitely generated R indecomposable module. The following assertions are equi¨alent.