Lattice isomorphisms of finite local rings

Associative rings are considered. By a lattice isomorphism, or projection, of a ring $R$ onto a ring $R^{\varphi}$ we mean an isomorphism $\varphi$ of the subring lattice $L(R)$ of $R$ onto the subring lattice $L(R^{\varphi})$ of $R^{\varphi}$. In this case $R^{\varphi}$ is called the projective image of a ring $R$ and $R$ is called the projective preimage of a ring $R^{\varphi}$. Let $R$ be a finite ring with identity and ${\rm Rad}\,R$ the Jacobson radical of $R$. A ring $R$ is said to be local if the factor ring $R/{\rm Rad}\,R$ is a field. We study lattice isomorphisms of finite local rings. It is proved that the projective image of a finite local ring which is distinct from $GF(p^{q^n})$ and has a nonprime residue ring is a finite local ring. For the case where both $R$ and $R^{\varphi}$ are local rings, we examine interrelationships between the properties of the rings.