On the Countability of Noetherian Dimension of Modules

Abstract

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ω^ω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.

Key Words:

• Noetherian dimension
• Krull dimension
• Locally Noetherian rings
• α-Atomic modules
• Finitely embedded modules

1991 Mathematics Subject Classification:

• Primary 16A53
• Secondary 16A56

Acknowledgments

We would like to thank the referee for many helpful remarks and for suggesting Example 1.10. The first author is partially supported by the Institute of Theoretical Physics and Mathematics (IPM).

Notes

^#Communicated by B. Huisgen-Zimmermann.