On type Krull dimension of modules

Document Type : Original Manuscript


1 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

2 Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran, Iran



In this paper, the concept of type Krull dimension of a module is introduced and some related properties are investigated.
Using this concept, we extend some basic results about modules with Krull dimension. It is shown that every module with homogeneous type Krull dimension has type Krull dimension equal to zero. Also, it is proved that an $R$-module $M$ has type Krull dimension if and only if it has type Noetherian dimension. We observe that, every module with Krull dimension has type Krull dimension, but its converse is not true in general. Further, we define t-Artinian (resp., t-Noetherian) modules and it is shown that if $M$ be a t-Artinian $R$-module with type Krull dimension, then it has Krull dimension and these two dimensions for $M$ coincide. At the end, we define the concept of $\alpha$-DICCT modules and it is proved that an $R$-module $M$ is $\alpha$-DICCT if and only if it has type Krull dimension.