On sums of diagonal and invertible formal matrices

This paper concerns properties of $k$-good formal matrix rings $K_n$ of order $n$ with rings $R_1, R_2, \dots, R_n$ on the main diagonal and $R_i-R_j$-bimodules $M_{ij}$ on other places. In the ring theory, various matrix rings play an important role. Above all I mean formal matrix rings. Formal matrix rings generalize a notion of matrix ring of order $n$ over a given ring. Every ring with nontrivial idempotents is isomorphic to some formal matrix ring. The endomorphism ring of a decomposable module also is a formal matrix ring. The studies of such rings are quite useful for solving some problems on endomorphism rings of Abelian groups. In this paper I show that every matrix form $K_n$ is the sum of diagonal matrix and invertible matrix. Also I give one condition when $K_n$ is the $k$-good ring.