On parallel Krull dimension of modules

Abstract

In this article, we introduce the concept of parallel Krull dimension of a module (briefly, p.Krull dimension), which is Krull-like dimension extension of the concept of DCC on poset of submodules parallel to itself. Using this concept, we extend some basic results about modules with this dimension. In particular, we show that an R-module M has Krull dimension if and only if it has homogeneous parallel Krull dimension with finite Goldie dimension and these two dimensions for M coincide. Furthermore, we define the concept of α-DICCP and we prove that M is an α-DICC module if and only if M is an α-DICCP module with finite Goldie dimension. Also, after defining of p-Artinian (resp., p-Noetherian) modules, we prove that if R is semiprime, p-Artinian commutative ring with finite Goldie dimension, then R is a Goldie ring.

Keywords:

• Parallel submodules
• atomic modules
• parallel Krull dimension
• Krull dimension

2020 Mathematics Subject Classification:

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

Acknowledgments

We would like to thank the referees for valuable comments and the careful reading of our manuscript.