The left (b,c)-core inverse in rings and its applications
Yicheng Zhang, Li Wang, Huihui Zhu · Filomat · 2025
Let R be a *-ring and let a, b, c ∈ R. The paper aims to introduce and investigate the left (b,c)-core inverses of a. The element a ∈ R is left (b,c)-core invertible if there exists some x ∈ R such that caxc = c, xcab = b and cax = (cax)* . Such an x is called a left (b, c)-core inverse of a. Several criteria for the left (b, c)-core inverse are given. Among of these, it is proved that a is left (b, c)-core invertible if and only if a is left (b, c)-invertible and c (or ca) is {1, 3}-invertible, under certain condition. Moreover, the connection between left (b, c)-core inverses and left (b, c)-inverses is established. Finally, y = xcax is the (b, c)-core inverse of a if and only if the descending chain caR ⊇ (ca)^{2} > yR ⊇ ...⊇ y^n R ⊇ ... stabilizes. Finally, the application of this type of generalized inverses is given to carbon emission.