Abstract:
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.
Keywords:finite local rings, lattice isomorphisms of associative rings.