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Reverse order law and forward order law for the (b, c)-Inverse
(International Linear Algebra Society, 2023) Višnjić, Jelena; Stanišev, Ivana; Ke, Yuanyuan
The reverse order law and the forward order law have been studied for various types of generalized inverses. The (b,c)-inverse is a generalization of some well known generalized inverses, such as the Moore-Penrose inverse, the Drazin inverse, the core inverse, etc. In this paper, the reverse order law for the (b,c)-inverse, in a unital ring, is investigated and an equivalent condition for this law to hold for the (b,c)-inverse is derived. Also, some known results on this topic are generalized. Furthermore, the forward order law for the (b,c)-inverse in a ring with a unity is introduced, for different choices of b and c. Moreover, as corollaries of obtained results, equivalent conditions for the reverse order law and the forward order law for the inverse along an element are derived.