On the small Krull dimension of modules

ABSTRACT

We introduce and study the concept of small Krull dimension of a module which is Krull-like dimension extension of the concept of DCC on small submodules. Using this concept we extend some of the basic results for modules with this dimension, which are almost similar to the basic properties of modules with Krull dimension. When for a module A with small Krull dimension, whose Rad(A) is quotient finite dimensional, then these two dimensions for Rad(A) coincide. In particular, we prove that if an R-module A has finite hollow dimension, then A has small Krull dimension if and only if it has Krull dimension. Consequently, we show that if A has properties AB5* and qfd, then A has s.Krull dimension if and only if A has Krull dimension.

KEYWORDS:

• Hollow dimension
• Krull dimension
• s.critical modules
• small submodules
• s.Krull dimension

2010 MATHEMATICS SUBJECT CLASSIFICATION:

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