Huihui Zhu

Institute of Mathematics of the Czech Academy of Sciences

Given a unital ring [ ] and [ ]. The element [ ] is called [ ]-polar along [ ] if there exists some [ ] such that [ ], [ ] and [ ]. Several properties of the [ ]-polarity along an element are given. Also, a new type of generalized inverses (i.e. the [ ]-inverse of [ ] along [ ]) is defined, and the connection between the [ ]-polarity along an element and the [ ]-inverse along an element is established. Finally, [ ]-JEP elements are introduced and investigated. Several results on the [ ]-EP element by Yang et al. [Comm. Algebra (2025)] are given as corollaries.

Our aim is to consider extensions of the systems for defining the 1D, D1, weak MPD and weak DMP inverses using an inner inverse, a minimal rank weak Drazin inverse and a minimal rank right weak Drazin inverse. As unique solutions of new extended systems, we introduce weak 1D and D1 inverses as wider classes of generalized inverses which contain the 1D, D1, inner-CEP, 1WG and inner-[ ]-WG inverses. We establish characterizations and formulae for weak 1D and D1 inverses. Applying the weak 1D and D1 inverses, solvability of some linear equations is proved. Consequently, a basic result about application of inner inverses for solving linear equations is obtained.