TSIU-KWEN LEELin J.-HQuynh T.C.2021-07-212021-07-21202149727https://www.scopus.com/inward/record.uri?eid=2-s2.0-85100172132&doi=10.1017%2fS0004972720001550&partnerID=40&md5=cf358dcf313aecdc9c33bf199c4df83ahttps://scholars.lib.ntu.edu.tw/handle/123456789/572146Let R be a semiprime ring with extended centroid C and let denote the set of all inner inverses of a regular element x in R. Given two regular elements in R, we characterise the existence of some such that. Precisely, if are regular elements of R and a and b are parallel summable with the parallel sum, then. Conversely, if for some, then is invariant for all, where is the smallest idempotent in C satisfying. This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. ['Invariance and parallel sums', Bull. Math. Sci. 10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings. ? 2021 Australian Mathematical Publishing Association Inc.journal article10.1017/S00049727200015502-s2.0-85100172132