Projections of Finite One Generated 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 else a lattice isomorphism) of the ring R onto the ring R' A ring R' is called the projective image of a ring R. Lattice isomorphisms of finite one generated rings with identity are studied. We elucidate the general structure of finite one generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one generated ring. We look at lattice isomorphisms of one generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m) where m 1 and n 1. We specify sufficient conditions for the projective image of a one generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.