Cyclically Presented Modules Over Rings of Finite Type

Abstract

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules.

Key Words:

• Cyclically presented modules
• Indecomposable injective modules
• Krull–Schmidt Theorem
• Semilocal rings

2000 Mathematics Subject Classification:

• 16L30
• 16D50
• 16E30

ACKNOWLEDGMENT

The research of the third author was partially supported by Ministero dell'Istruzione, dell'Università e della Ricerca, Italy (Prin 2007 “Rings, algebras, modules and categories”) and by Università di Padova (Progetto di Ricerca di Ateneo CPDA071244/07).

Notes

Communicated by T. Albu.