The theory of generalized inverses has its roots both on semigroup theory and on matrix and operator theory. In this seminar we will focus on the study of the generalized inverse of von Neumann, group, Drazin and Moore-Penrose in a purely algebraic setting. We will present some recent results dealing with the generalized inverse of certain types of matrices over rings, emphasizing the proof techniques used. Namely, we will address recent results related to the existence of the group inverse of 2 × 2 block matrices with a zero (2,2) block in terms of its blocks. This problem is called the 220 group inverse problem. These results have been very recently extended to the case the (2,2) block has a group inverse, not being necessarily zero. We will consider 2 × 2 block matrices over a general (not necessarily von Neumann regular) ring, assuming some local regularity on the elements. We will use outer inverses and the Brown-McCoy shift to characterize the existence of the inverse and group inverse of such block matrices. Furthermore, we will present recent results on this problem with respect to the Moore-Penrose inverse.
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