Modules with chain condition on non-finitely generated submodules. (English) Zbl 1384.16015
Summary: 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.
MSC:
| 16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
Keywords:
non-finitely generated module; quotient finite dimensional module; Krull dimension; Noetherian dimensionReferences:
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