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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.
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