Vol. 69, No. 2, 1977

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Modules whose quotients have finite Goldie dimension

Victor P. Camillo

Vol. 69 (1977), No. 2, 337–338

If M is a module and M is a submodule of M, then N is irreducible in M if N cannot be written as a proper intersection of two submodules of M. The purpose of this note is to study modules whose submodules can be written as a finite intersection of irreducible submodules. Such modules are characterized by the fact that their quotients all have finite Goldie dimension, so they are called q.f.d. modules.

The main result is: A module M is q.f.d. if and only if every submodule N has a finitely generated submodule T such that N∕T has no maximal submodules. Because T is finitely generated this generalizes a theorem of Shock (using his ideas), who showed a q.f.d. module M having the property that every subquotient of M has a maximal submodule must be noetherian (and conversely, of course).

Mathematical Subject Classification
Primary: 16A46, 16A46
Received: 22 July 1976
Published: 1 April 1977
Victor P. Camillo