The paper consists of some interesting results such as proving that “the projective image of a finite local ring which is distinct from the field GF\((p^{q^n})\) and has a non-prime residue field is a finite local ring”.
Author’s abstract: “Associative rings are considered. By a lattice isomorphism, or projection, of a ring \(R\) onto a ring \(R^\phi\) we mean an isomorphism \(\phi\) of the subring lattice \(L(R)\) of \(R\) onto the subring lattice \(L(R^\phi)\) of \(R^\phi \). In this case \(R^\phi\) is called the projective image of a ring \(R\) and \(R\) is called the projective preimage of a ring \(R^\phi \). Let \(R\) be a finite ring with identity and \(\operatorname{Rad} R\) the Jacobson radical of \(R\). A ring \(R\) is said to be local if the factor ring \(R/\operatorname{Rad} R\) 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 field is a finite local ring. For the case where both \(R\) and \(R^\phi\) are local rings, we examine interrelationships between the properties of the rings.”