Projections of finite commutative rings with identity

Associative rings $R$ and $R'$ are said to be lattice-isomorphic if their subring lattices $L(R)$ and $L(R')$ are isomorphic. An isomorphism of the lattice $L(R)$ onto the lattice $L(R')$ is called a projection (or a lattice isomorphism) of the ring $R$ onto the ring $R'$. A ring $R'$ is called the projective image of a ring $R$. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring $GR(p^n,m)$, where $n>1$ and $m>1$) leads to strong connections between the properties of $R$ and $R'$.