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.

For a left module $_{R}M$ over a non-commutative ring $R$, we define the concept of a strongly semicommutative module as a generalization of the reduced module. This notion constitutes a distinct and stronger category within the class of semicommutative modules. We demonstrate that a module $_{R}M$ is strongly semicommutative if and only if $_{A_{n}(R)}A_{n}(M)$ is strongly semicommutative. Additionally, we establish that $_{R}M$ is strongly semicommutative if and only if $_{R[x]}M[x]$ is strongly semicommutative; this is also equivalent to $_{R[x, x^{-1}]}M[x, x^{-1}]$ being strongly semicommutative. Among our findings, we prove that if $_{R}M$ is strongly semicommutative, then for any reduced submodule $N$ of $M$, the quotient module $M/N$ is also strongly semicommutative. We provide examples of semicommutative modules that are not strongly semicommutative and show that the class of strongly semicommutative modules remains closed under localization.

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.