On Nilpotent Elements and Armendariz Modules

Nazeer Ansari, Kholood Alnefaie, Shakir Ali (), Adnan Abbasi and Kh. Herachandra Singh
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Nazeer Ansari: Department of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle 517325, Andhra Pradesh, India
Kholood Alnefaie: Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
Shakir Ali: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, Uttar Pradesh, India
Adnan Abbasi: School of Advances Sciences and Languages, VIT Bhopal University, Kothrikalan, Sehore 466114, Madhya Pradesh, India
Kh. Herachandra Singh: Department of Mathematics, Manipur University, Canchipur, Imphal 795003, Manipur, India

Mathematics, 2024, vol. 12, issue 19, 1-13

Abstract: For a left module M R over a non-commutative ring R , the notion for the class of nilpotent elements ( n i l R ( M ) ) was first introduced and studied by Sevviiri and Groenewald in 2014 ( Commun. Algebra , 42 , 571–577). Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and n i l R ( 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 M R . Additionally, we establish that the matrix module M n ( M ) is nil-Armendariz over M n ( R ) and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.

Keywords: nilpotent element; Armendariz module; Armendariz ring; nil-Armendariz module (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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