On Α -QUASI SHORT MODULES

Abstract

Abstract. We introduce and study the concept of α-quasi short modules. Using this concept we extend some of the basic results of α-short modules to α-quasi short modules. We observe that if M is an α-quasi short module then the Noetherian dimension of M is α or α + 1 or α + 2.Mathematics Subject Classification (2010): 16P60, 16P20, 16P40

ON \alpha-ALMOST QUASI ARTINIAN MODULES

“…Remark 3.8. Since our results in this article are related to the results in [14,21] and there are two minor errors in these referencess (one in each), I strongly recommend the reader to see [15,22] for corrections.…”

“…In particular, we observe that each α-almost quasi Artinian module M has quasi Krull dimension and qk-dim M ≤ α. We also introduce and study the concept of α-quasi Krull modules, which is the dual of α-quasi short modules, see [6]. We recall that an R-module M is called an α-Krull module, if for each submodule N of M , either k-dim N ≤ α or k-dim M N ≤ α and α is the least ordinal number with this property.…”

“…We continue with our definition of α-almost quasi Artinian modules. We continue with the following definition, which is in fact the dual of α-quasi short modules, see [6,Definition 2.7], and in the subsequent results we try to present counterparts of the appropriate results in [6]. We need the following result.…”

“…Davoudian, Karamzadeh and Shirali in [14] introduce and study the concepts of α-short modules and α-almost Noetherian modules. We recall that an R-module M is called an α-short module, if for each submodule N of M , either n-dim N ≤ α or n-dim M N ≤ α and α is the least ordinal number with this property.…”

“…We recall that an R-module M is called an α-short module, if for each submodule N of M , either n-dim N ≤ α or n-dim M N ≤ α and α is the least ordinal number with this property. We also recall that an R-module M is called α-almost Noetherian, if for each proper submodule N of M , n-dim N < α and α is the least ordinal number with this property, see [14]. Later Davoudian, Halali and Shirali undertook a systematic study of the concepts of α-almost Artinian and α-Krull modules, which are the dual of the concepts of α-almost Noetherian and α-short modules, respectivly, see [12].…”