Projections of finite rings

Let $R$ and $R^{\varphi}$ be associative rings with isomorphic subring lattices, and $\varphi$ be a lattice isomorphism (or else a projection) of the ring $R$ onto the ring $R^{\varphi}$. We call $R^{\varphi}$ the projective image of a ring $R$ and call $R$ itself the projective preimage of a ring $R^{\varphi}$. The main result of the first part of the paper is Theorem 5, which proves that the projective image $R^{\varphi}$ of a one-generated finite $p$-ring $R$ is also one-generated if $R^{\varphi}$ at the same time is itself a $p$-ring. In the second part, we continue studying projections of matrix rings. The main result of this part is Theorems 6 and 7, which prove that if $R=M_n(K)$ is the ring of all square matrices of order $n$ over a finite ring $K$ with identity, and $\varphi$ is a projection of the ring $R$ onto the ring $R^{\varphi}$, then $R^{\varphi}=M_n(K')$, where $K'$ is a ring with identity, lattice-isomorphic to the ring $K$.