Reveals new properties of partial isometries in rings with involution, suggesting fresh insights into their behavior.
In this paper, we discover some properties of partial isometries by some related idempotent elements, projections, PE elements, mainly via constructing some equations to study new 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.
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Wang et al. (2025) studied this question.
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