Reverse Order Law for the Core Inverse in Rings

Abstract

In this paper, necessary and sufficient conditions of the one-sided reverse order law (ab) # = b # a # , the two-sided reverse order law (ab) # = b # a # and (ba) # = a # b # for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as (ab) # = b # (abb #) # , a # = b(ab) # and (ab) # = b # a # , are also considered.

Inner-gMP and gMP-inner inverses

“…Consequently, for A ∈ B(H) # , the core-EP (or *core-EP) inverse of A coincides with the core (or dual core) inverse [1]. Significant results about the core and core-EP inverse are established in [2,5,6,9,11,[21][22][23].…”

Further results on weighted core inverse in a ring

“…The answer to this query was first studied in [16] with a few necessary and sufficient conditions on the reverse order law for the Moore-Penrose inverse in from of rectangular matrices [30,29]. From that time, reverse order law for generalized inverses have significantly impacted many areas of science and engineering in the context of matrices [4], operators [10], tensors [24,27] and elements of rings with involution [34,23]. In particular, Koliha et al [18] discussed the reverse order law for the MoorePenrose invertible elements and Liu et al [20] derived some equivalences of the reverse order law for the group invertible elements in a ring.…”

“…In particular, Koliha et al [18] discussed the reverse order law for the MoorePenrose invertible elements and Liu et al [20] derived some equivalences of the reverse order law for the group invertible elements in a ring. Further, the authors of [34] have discussed a few one-sided reverse order law and two-sided reverse order law for the core inverse in rings. The vast work on the reverse order law [22,11,30] focuses our attention to discuss reverse order laws for weighted core and dual inverse in a ring.…”

A 3D biophysical model for cancer spheroid cell-enhanced invasion in collagen-oriented fiber microenvironment*

“…[12] Through 3D cell culture, the morphology, function, and characteristics of tumor tissue in vivo can be simulated in vitro, such as cell-cell interaction, cell migration, cell signaling, drug penetration, response, and drug resistance. [13,14] At present, commonly used methods for generating tumor spheroids include: hanging drop, rotating flask, rotating cell culture system, ultra-low attachment plate, and microfluidic.…”