Armendariz and semicommutative rings are generalizations of reduced rings. In [4], 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.
