Weighted w-core inverses in rings

Authors：Wu, Liyun; Zhu, Huihui^*

Source：CZECHOSLOVAK MATHEMATICAL JOURNAL, 2023, 73(2): 581-602.

DOI：10.21136/CMJ.2022.0134-22

• inverse along an element
• {e
• 1
• 3}-inverse
• {f
• 4}-inverse
• weighted Moore-Penrose inverse
• (v
• w)-(b
• c)-inverse
• w-core inverse
• dual v-core inverse

Let R be a unital *-ring. For any a, s, t, v, w is an element of R we define the weighted w-core inverse and the weighted dual s-core inverse, extending the w-core inverse and the dual s-core inverse, respectively. An element a is an element of R has a weighted w-core inverse with the weight v if there exists some x is an element of R such that awxvx = x, xvawa = a and (awx)* = awx. Dually, an element a is an element of R has a weighted dual s-core inverse with the weight t if there exists some y is an element of R such that ytysa = y, asaty = a and (ysa)* = ysa. Several characterizations of weighted w-core invertible and weighted dual s-core invertible elements are given when weights v and t are invertible Hermitian elements. Also, the relations among the weighted w-core inverse, the weighted dual s-core inverse, the e-core inverse, the dual f-core inverse, the weighted Moore-Penrose inverse and the (v, w)-(b, c)-inverse are considered.