Work

On Nilpotent Elements and Armendariz Modules

Year: 2024

Type: article

Abstract: For a left module RM over non-commutative ring R, the notion for the class of nilpotent elements (nilR(M)) was first introduced and studied by SSevviiri and Groenewald . Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and nilR(M)=0 in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module M/N is nil-Armendariz if and only if N is within the nilpotent class of RM. Additionally, we establish that the matrix module Mn(M) is nil-Armendariz over Mn(R) and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.

Source: Mathematics

DOI: 10.3390/math12193133

Authors: Nazeer Ansari, Kholood Alnefaie, Shakir Ali, Adnan Abbasi, Kh. Herachandra Singh

Institutions: Taibah University, Aligarh Muslim University, VIT Bhopal University, Manipur University

Language: English

Topic: Rings, Modules, and Algebras

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,955

Funder: Taibah University