About $k$-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.