Generalized Cline's Formula and Jacobson's Lemma in a Ring

We present a new generalized version of Cline's formula and Jacobson's lemma for g-Drazin inverses in a ring. These generalized results extend many known results, e.g., Chen and Abdolyousefi [Generalized Jacobson's lemma in a Banach algebra, Comm. Algebra 49 (2021) 3263–3272], and Yan and Zeng [The generalized inverses of the products of two elements in a ring, Turkish J. Math. 44 (2020) 1744–1756].