Abstract
In this article, we study modules with chain condition on non-finitely generated submodules. We show that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then M has Noetherian dimension and its Noetherian dimension is less than or equal to one. In particular, we observe that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then every submodule of M is countably generated. We investigate that if an R-module M satisfies the descending chain condition on non-finitely generated submodules, then M has Krull dimension and its Krull dimension may be any ordinal number $$\alpha $$ . In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.
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