On semicommutativity of rings relative to hypercenter
Nazeer Ansari, Kh. Herachandra singh
TL;DR
The article introduces $\mathscr{H}$-semicommutative rings, a hypercenter-based generalization of semicommutativity defined by $uv=0 \Rightarrow uRv \subseteq T(R)$ for all $r\in R$, where $T(R)$ is the hypercenter. It situates this class strictly between Zero-Insertive rings (ZI) and Abelian rings and proves closure under subrings, with nil rings automatically satisfying $\mathscr{H}$-semicommutativity. It proves that $Nil(R)$ is an ideal, that $R$ is $2$-primal, and that constructions such as the matrix subrings $S'_{n}(R)$, localization, and polynomial extensions preserve the property, while connecting to Baer, quasi-Baer, $p.p.$, and $p.q.$ notions. It further shows that under additional conditions, $\mathscr{H}$-semicommutativity implies strong regularity or nil-singularity, and provides examples illustrating both inclusion beyond reduced/central-semicommutative rings and noninheritance to certain quotients. These results extend the theory of generalized semicommutativity and link it to several classical ring classes and constructions.
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.