On Nilpotent Elements and Armendariz Modules

For a left module ${}_{R}M$ over a non-commutative ring R, the notion for the class of nilpotent elements ($ni{l}_R\left(M\right)$) 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 $ni{l}_R\left(M\right)=0$ 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 ${}_{R}M$. Additionally, we establish that the matrix module $M_n\left(M\right)$ is nil-Armendariz over $M_n\left(R\right)$ and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.