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Let \( R \) be a ring with 1. Jacobson's lemma states that for any two elements \( a, b \in R \), the element \( 1 - ab \) is invertible if and only if the element \( 1 - ba \) is invertible (for a humorous "proof"of this, see [1]). There are several generalizations of the concept of "inverse", starting from say, von Neumann regular rings all the way up to group inverses, Moore-Penrose inverses, Drazin inverses etc (see pages 9, 15 in [2] for more details). Now, it is only natural to ask for the validity or otherwise of Jacobson's lemma for rings with these new types of inverses. In this talk, I will mention a few examples of such theorems known from the literature (see, e.g. [3]).

1. Halmos, P, R. Does mathematics have elements? Math. Intelligencer, 3 (1981) 147-153, (MR642132, 83e:00004)
2. Kirkland, S.J. and Newmann, M. Group inverses of matrices and their applications, CRC Press, New York, 2013.
3. Corach, Duggal, Harte. Extensions of Jacobson's Lemma, Communications in Algebra, 41 (2013), 520-531.