Vol. 19, No. 1, 1966

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ISSN: 0030-8730
On indecomposable modules over rings with minimum condition

Robert Ray Colby

Vol. 19 (1966), No. 1, 23–33
Abstract

Let A be an associative ring with left minimum condition and identity. Let g(d) denote the number of nonisomorphic indecomposable A-modules which have composition length d, d a nonnegative integer. If, for each integer n, there exists an integer d > n, such that g(d) = , A is said to be of strongly unbounded module type.

Assume that the center of the endomorphism ring of each simple (left) A-module is infinite. The following results concerning the structure of rings of strongly unbounded type are obtained.

I. If the ideal lattice of A is infinite, then A is of strongly unbounded module type.

II. If A is commutative, then A has only a finite number of (nonisomorphic) finitely generated indecomposable modules if and only if the ideal lattice of A is distributive. Otherwise, A is of strongly unbounded module type.

III. If the ideal lattice of A contains a vertex V of order greater than three such that, for some primitive idempotent e A, the image V e of V is a vertex of order greater than three in the submodule lattice of Ae, then A is of strongly unbounded module type.

These results are generalizations of earlier ones obtained by J. P. Jans for finite dimensional algebras over algebraically closed fields.

Mathematical Subject Classification
Primary: 16.40
Milestones
Received: 30 July 1965
Published: 1 October 1966
Authors
Robert Ray Colby