Lattice Isomorphisms of Finite Local Rings

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