Left w-Core Inverses in Rings with Involution

作者：Zhu, Huihui; Wang, Chengcheng; Wang, Qing-Wen^*

来源：Mediterranean Journal of Mathematics, 2023, 20(6): 337.

DOI：10.1007/s00009-023-02541-9

• w-core inverses
• right w-core inverses
• left inverses along an element
• {1,3}-inverses
• Moore-Penrose inverses

In Zhu et al. (Linear Multilinear Algebra 71:528-544, 2023), the authors described the left w -core inverse by principal ideals in *-ring, and asked whether it can be defined by the solution of equations. In this paper, we answer the question in the positive. For any *-ring R and a, w is an element of R, the element a is called left w -core invertible if there is some x is an element of R satisfying awxa = a, xawa = a and (awx)& lowast; = awx. Several criteria for left w -core inverses are presented. Among of these, it is proved that a is left w -core invertible if and only if w is left invertible along a, a (or aw) is {1, 3}-invertible and a is an element of awR. Also, the relations among left w -core inverses, w -core inverses, and other generalized inverses are established. As applications, several characterizations for the Moore- Penrose inverse, the core inverse, and the pseudo-core inverse are given.

• 单位
  上海大学