Projections of Galois rings

Let $R$ and $R^\varphi$ be associative rings with isomorphic subring lattices and $\varphi$ be a lattice isomorphism (a projection) of the ring $R$ onto the ring $R^\varphi$. We call $R^\varphi$ the projective image of a ring $R$ and call the ring $R$ itself the projective preimage of a ring $R^\varphi$. 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 $K=Z/p^nZ$, $p$ 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:
THEOREM. Let $R=GR(p^n,q^m)$, where $n>1$ and $m>1$. Then $R^\varphi\cong R$.