It is well known that a square matrix with all its columns linearly independent has a unique inverse G satisfying AG=GA=I. However, if A is not square or if columns of A are not linearly independent, then such an inverse does not exist. Moore and Penrose independently generalized the notion of inverses to singular matrices. This led to the theory of generalized inverses. In the course of time different generalized inverses like group inverse, Drazin inverse, Core inverse etc., were introduced. The present author introduced Core-EP inverse in 2013. The author has also worked on the theory of generalized inverse of elements of rings. Many of the notions defined in the case of matrices have been extended for elements of rings and their properties have been studied. A few of the research problems the author has worked on is characterization of outer inverses in semigroups, bordering method to find the generalized inverses, reverse order law for generalized inverses, inverse complements, core-nilpotent decomposition etc.,
- DRAZIN INVERSE AND GENERALIZATION OF CORE-NILPOTENT DECOMPOSITION | Journal of Algebra and Its Applications (worldscientific.com)
Prasad, K. Manjunatha, "Generalized Inverses of Elements of Rings and Matrices over Rings" (2022). Interdisciplinary Collection. 14.