TSIU-KWEN LEELin J.-H.2021-07-212021-07-212021333883https://www.scopus.com/inward/record.uri?eid=2-s2.0-85100134619&doi=10.5486%2fPMD.2021.8842&partnerID=40&md5=735b1df58e0234c2b84761642d8bbe3ehttps://scholars.lib.ntu.edu.tw/handle/123456789/572141In the paper, we obtain an explicit formula for the outer inverses of a regular element in an arbitrary ring. It becomes calculable for outer inverses. We characterize the triplet ba?c (resp. ba+c ) invariant under all inner inverses a? (resp. reflexive inverses a+) of a in a semiprime ring. It is also proved that if R is a regular ring and a, b, c ? R, then the triplet b?c is invariant under all outer inverses ? of a if and only if E[a] E[b] E[c] = 0. Here, for x ? R, E[x] is the smallest idempotent in the extended centroid of R such that x = E[x]x. These answer two questions in Hartwig and Patr?cio [12]. ? 2021 University of Debrecen, Institute of Mathematics. All rights reserved.journal article10.5486/PMD.2021.88422-s2.0-85100134619