Projections of Galois Rings

Let R and R (phi) be associative rings with isomorphic subring lattices and phi be a lattice isomorphism (a projection) of the ring R onto the ring R (phi) . We call R (phi) the projective image of a ring R and call the ring R itself the projective preimage of a ring R (phi) . We study lattice isomorphisms of Galois rings. By a Galois ring we mean a ring GR(p (n) , m) isomorphic to the factor ring K[x]/(f(x)), where K = Z/p (n) Z, p is a prime, f(x) is a polynomial of degree m irreducible over K, and (f(x)) is a principal ideal generated by the polynomial f(x) in the ring K[x]. Properties of the lattice of subrings of a Galois ring depend on values of numbers n and m. A subring lattice L of GR(p (n) , m) has the simplest structure for m = 1 (L is a chain) and for n = 1 (L is distributive). It turned out that only in these cases there are examples of projections of Galois ring onto rings that are not Galois rings. We prove the following result (Thm. 4). Let R = GR(p (n) , q (m) ), where n > 1 and m > 1. Then R (phi) a parts per thousand... R.