About fc-nil-good formal matrix rings

In 2018, Abdolyusefi, Ashrafi, and Chen gave a definition of a 2-nil-good ring element in their work, generalizing the notion of a graceful ring element introduced two years earlier by Kalugeryan and Lam, as well as the definition of a 2-nil-good ring. In the same work, it was shown that the Morita context ring, i.e. a formal matrix ring of the second order is 2-nil-good if the rings over which it is considered are themselves 2-nil-good. In this paper, we generalize further, defining k-nil-good elements and k-nil-good rings, and state a condition under which a formal matrix ring of an arbitrary finite order is k-nil-good.