Direct Sums of Representations as Modules over Their Endomorphism Rings
✍ Scribed by Birge Huisgen-Zimmermann; Manuel Saorı́n
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 178 KB
- Volume
- 250
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Our investigation into the endo-structure of infinite direct sums i∈ I M i of indecomposable modules M i -over a ring R with identity-is centered on the following question: If S = End R i∈ I M i , how much pressure, in terms of the S-structure of i∈ I M i , is required to force the M i into finitely many isomorphism classes? One of the consequences of our principal result in this direction (Theorem H of Section 4) is as follows. If all of the M i are endofinite (think, for instance, of finitely generated or generic modules over an Artin algebra) and if M t t∈T is a transversal of the isomorphism types of the M i , then the following conditions (1)-( 4) are equivalent: (1) T is finite; (2) i∈ I M i is endo-artinian and M t t∈T