Projections of finite nonnilpotent rings

Associative rings $R$ and $R'$ are said to be lattice-isomorphic if their subring lattices $L(R)$ and $L(R')$ are isomorphic. An isomorphism of the lattice $L(R)$ onto the lattice $L(R')$ is called a projection (or lattice isomorphism) of the ring $R$ onto the ring $R'$. A ring $R'$ is called a projective image of a ring $R$. Whenever a lattice isomorphism $\varphi$ implies an isomorphism between $R$ and $R^\varphi$, we say theat the ring $R$ is determined by its subring lattice. The present paper is a continuation of previous research on lattice isomorphisms of finite rings. We give a complete description of projective images of prime and semiprime finite rings. One of the basic results is the theorem on lattice definability of a matrix ring over an arbitrary Galois ring. Projective images of finite rings decomposable into direct sums of matrix rings over Galois rings of different types are described.