$k$-good formal matrix rings of infinite order

Let $k$ be an integer that is greater than or equal to $2$. The ring $R$ is said to be $k$-good if every element of $R$ is the sum of $k$ invertible elements of $R$. We have showed that the ring of formal row-finite matrices will be $k$-good if all rings from its main diagonal are $k$-good. Also some applications of this result are given, particularly to the problem of $k$-goodness of the ring of endomorphisms of decomposable module or Abelian group.