Projections of semilocal 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 a ring $R$ onto the subring lattice $L(R^{\varphi})$ of a ring $R^{\varphi}$. Let $M_n(GF(p^k))$ be the ring of all square matrices of order $n$ over a finite field $GF(p^k)$, where $n$ and $k$ are natural numbers, $p$ is a prime. A finite ring $R$ with identity is called a semilocal (primary) ring if $R/{\rm Rad} R\cong M_n(GF(p^k))$. It is known that a finite ring $R$ with identity is a semilocal ring iff $R\cong M_n(K)$ and $K$ is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if $\varphi$ is a projection of a ring $R=M_n(K)$, where $K$ is an arbitrary finite local ring, onto a ring $R^{\varphi}$, then $R^{\varphi}=M_n(K')$, in which case $K'$ is a local ring lattice-isomorphic to the ring $K$. We thus prove that the class of semilocal rings is lattice definable.