On the countably Noetherian dimension of modules

Abstract

We introduce and study the c.Noetherian dimension of an $R-$module A, denoted by $cn-\mathit{dim}(A),$ which is the Noetherian dimension of the partially ordered set (briefly, poset) of countably generated submodules of A and prove that A has Noetherian dimension if and only if it has c.Noetherian dimension and furthermore $cn-\mathit{dim}(A)\le n-\mathit{dim}(A)\le cn-\mathit{dim}(A)+1.$

Keywords:

• Noetherian dimension
• c.g. submodule
• c.Noetherian dimension
• q.f.d. module

2020 Mathematics Subject Classification:

• Primary: 16P40
• 16P60
• Secondary: 16P20

Acknowledgments

The author is deeply grateful to the referee for his/her meticulous reading and valuable suggestions which have definitely improved the paper.