Version 1
: Received: 2 September 2024 / Approved: 2 September 2024 / Online: 3 September 2024 (07:55:50 CEST)
How to cite:
Ansari, N.; Alnefaie, K.; Ali, S.; Abbasi, A.; Singh, K. H. On Nilpotent Elements and Armendariz Modules. Preprints2024, 2024090181. https://doi.org/10.20944/preprints202409.0181.v1
Ansari, N.; Alnefaie, K.; Ali, S.; Abbasi, A.; Singh, K. H. On Nilpotent Elements and Armendariz Modules. Preprints 2024, 2024090181. https://doi.org/10.20944/preprints202409.0181.v1
Ansari, N.; Alnefaie, K.; Ali, S.; Abbasi, A.; Singh, K. H. On Nilpotent Elements and Armendariz Modules. Preprints2024, 2024090181. https://doi.org/10.20944/preprints202409.0181.v1
APA Style
Ansari, N., Alnefaie, K., Ali, S., Abbasi, A., & Singh, K. H. (2024). On Nilpotent Elements and Armendariz Modules. Preprints. https://doi.org/10.20944/preprints202409.0181.v1
Chicago/Turabian Style
Ansari, N., Adnan Abbasi and Kh. Herachandra Singh. 2024 "On Nilpotent Elements and Armendariz Modules" Preprints. https://doi.org/10.20944/preprints202409.0181.v1
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.
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
