Generalizations of partial isometries, EP and normal elements in rings
Honglin Zou, Jianlong Chen, Xiaofeng Chen · Filomat · 2024
Let a, d be two elements in rings and a?d be the inverse of a along d. Some characterizations of the equality a?d = d are given in rings. We also investigate the equivalent conditions for aa?d = a?da to hold, as well as ad = da. We prove that a?d = d if and only if ad is idempotent, when a ? R?d; aa?d = a?da if and only if there exists t ? R?1 such that a?d = at = ta, when a ? R??d; ad = da if and only if a?d(a + d) = (a + d)a?d, when a ? R??d. Thus, some well-known results on partial isometries, EP and normal elements in rings are extended to more general settings.