Modules with chain condition on uncountably generated submoduled

Abstract

In this paper, we study modules with chain condition on uncountably generated submodules. We show that if an [Formula: see text]-module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then its Goldie dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable cardinal number. We also show that if a quotient finite dimensional module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then it has Noetherian dimension and its Noetherian dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable ordinal number. We also investigate that if a quotient finite dimensional module [Formula: see text] satisfies the descending chain condition on uncountably generated submodules, then [Formula: see text] has Krull dimension and its Krull dimension may be any ordinal number [Formula: see text].