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 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.

Armendariz and semicommutative rings are generalizations of reduced rings. In \cite{IN}, I.N. Herstein introduced the notion of a hypercenter of a ring to generalize the center subclass. For a ring $R$, an element $a \in R$ is called hypercentral if $ax^{n}=x^{n}a$ for all $x \in R$ and for some $n=n(x,a) \in \mathbb{N}$. Motivated by this definition, we introduce $\mathscr{H}$-Semicommutative rings as a generalization of semicommutative rings and investigate their relations with other classes of rings. We have proven that the class of $\mathscr{H}$-Semicommutative rings lies strictly between Zero-Insertive rings (ZI) and Abelian rings. Additionally, we have demonstrated that if $R$ is $\mathscr{H}$-semicommutative, then for any $n \in \mathbb{N}$, the matrix subring $S_{n}^{'}(R)$ is also $\mathscr{H}$-semicommutative. Among other significant results, we have established that if $R$ is $\mathscr{H}$-semicommutative and left $SF$, then $R$ is strongly regular. We have also shown that $\mathscr{H}$-semicommutative rings are 2-primal, providing sufficient conditions for a ring $R$ to be nil-singular. Additionally, we have proven that if every simple singular module over $R$ is wnil-injective and $R$ is $\mathscr{H}$-semicommutative, then $R$ is reduced. Furthermore, we have studied the relationship of $\mathscr{H}$-semicommutative rings with the classes of Baer, Quasi-Baer, p.p. rings, and p.q. rings in this article, and we have provided some more relevant results.

In this paper, we introduce the concept of nil-semicommutative modules and present it as an extension of nil-semicommutative rings to modules. We prove that the class of nil-semicommutative modules is contained in the class of weakly semicommutative modules, while that of the converse may not be true. We also show that in case of semicommutative modules and nil-semicommutative modules, one does not imply the other. Moreover, for a given nil-semicommutative ring, we provide the conditions under which the same can be extended to a nil-semicommutative module and show that the converse may not be true in general. We also find the conditions under which the quotient module M/N is nil-semicommutative if and only if M is nil-semicommutative. Further, we also prove that for a left R-module M, _RM is nil-semicommutative iff its localization S^{-1}M over the ring S^{-1}R is also nil-semicommutative. Lastly, we explore whether the nil-semicommutative property is preserved under direct product of modules.