Modules with Chain Condition on Non-finitely Generated Submodules

作者：Davoudian Maryam^*

来源：Mediterranean Journal of Mathematics, 2018, 15(1): UNSP 1.

DOI：10.1007/s00009-017-1047-y

• Non-finitely generated module
• quotient finite dimensional module
• Krull dimension
• Noetherian dimension

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 a. In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.

• 出版日期2018-2