On \(C \sqrt{J}\) rings

Abstract

A ring \(R\) in which every \(x\in R\) is expressible as addition of \(c\in C(R)\) and an element \(z\in \sqrt{J(R)}\), where \(\sqrt{J(R)} = \{ z : z^n \in J(R) \mathrm{ \ for \ some \ n \geq 1}\}\) is called a \(C\sqrt{J}\) ring. The class of \(C\sqrt{J}\) rings is a proper subclass of the class of \(CU\) rings and includes \(CN\) and \(CJ\) rings properly. After developing the basic properties of \(C\sqrt{J}\) rings, we connect them with several other rings. We examine how this concept behaves across different extensions like matrices, Dorroh extension, \(T[R,S]\), and \(T(R,M)\). We investigate the condition to characterize the group rings as a \(C\sqrt{J}\) ring.