The Noetherian dimension of modules versus their α-small shortness

Document Type : Research Paper

Author

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

Abstract

In this article, we first consider concept of small Noetherian dimension of a module, which is dual to the small krull dimension, denoted by sn−dimA, and defined to be the codeviation of the poset of the small submodules of A. We prove that if an R-module A with finite hollow dimension, has small Noetherian dimension, then A has Noetherian dimension and sn−dimA≤n−dimA≤sn−dimA+1. Last we introduce the concept of α-small short modules, i.e., for each small submodule S of A, either n−dimS⩽α or sn−dimAS⩽α and α is the least ordinal number with this property and by using this concept, we extend some of the basic results of short modules to α-small short modules. In particular, we prove that if A is an α-small short module, then it has small Noetherian dimension and sn−dimA=α or sn−dimA=α+1. Consequently, we show that if A is an α-small short module with finite hollow dimension, then α≤n−dimA≤α+2.

Keywords

References

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Algebraic Structures and Their Applications
Volume 10, Issue 1
February 2023
Pages 1-15

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