The characterizations of partial isometries in a ring with involution

Abstract

In this paper, we discover some properties of partial isometries by some related idempotent elements, projections, PE elements, mainly via constructing some equations to studynew characterizations of PI elements in a rings with involution. The paper explores the representations of the general solution of univariate and bivariate equations, and then combining PI elements, it further optimizes the form of solutions to characterize PI elements. This work obtains some necessary and sufficient conditions of PI elements, and its new characterizations enrich the understanding of partial isometries.