Good formal matrix rings over residue class rings

For an arbitrary prime p ring E((Z/p(2)Z) circle plus (Z/pZ)) is a semilocal ring with p(5) elements that cannot be embedded in any matrix ring over commutative ring. In a more general case - a ring E((Z/p(m)Z) circle plus (Z/p(n)Z)), m. n, is isomorphic to a formal matrix ring ( [GRAPHICS] ) There are cryptographic systems based on the arithmetic of E((Z/p(2)Z) circle plus (Z/pZ)). We show that ring E((Z/p(m)Z) circle plus (Z/p(n)Z)) is 2-good and 2-nil-good for p > 2 and not good for p = 2 and m > n. Theorem 3.3. Let p be a prime and p > 2, m >= n, then E((Z/p(m)Z) circle plus (Z/p(n)Z)) is a 2-good ring. What if p = 2? In case of m = n, we have E((Z/(2)nZ) circle plus (Z/(2)nZ)) = M(2, Z/2(n)Z) which is 2-good. Theorem 3.5. Let m > n, then for a matrix A = [GRAPHICS] is an element of E((Zz/2(m)Z) circle plus (Z / 2(m) Z)), a, b, c, d. Z, the following statements are true: 1) Matrix A is 2-good if a and d are even; 2) Matrix A is 3-good if a and d are odd; 3) Matrix A is not good if a and d are numbers of different parity. Thus, formal matrix ring E((Z/2(m)Z) circle plus (Z/2(n)Z)), m > n, is not good.