On a class of 3-good formal matrix rings

We study formal matrix rings with values in a given ring and with a matrix of factors consisting of 0 and 1. Under the indicated restrictions, the formal matrix ring can be represented as a splitting extension of one of its nilpotent ideal using the product of ordinary matrix rings, and the question of the invertibility of the formal matrix is reduced to to the question of the invertibility of ordinary matrices over a ring. Under some additional conditions imposed on the matrix of factors, it is possible to use the well-known Henriksen theorem and prove that every element of a formal matrix ring is the sum of three invertible elements of this ring. Finally, we give examples of such formal matrix rings of orders 4 and 5.