On semicommutativity of rings relative to hypercenter

Abstract

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.