Projections of Galois Rings

Abstract

Let R and R (phi) be associative rings with isomorphic subring lattices and phi be a lattice isomorphism (a projection) of the ring R onto the ring R (phi) . We call R (phi) the projective image of a ring R and call the ring R itself the projective preimage of a ring R (phi) . We study lattice isomorphisms of Galois rings. By a Galois ring we mean a ring GR(p (n) , m) isomorphic to the factor ring K[x]/(f(x)), where K = Z/p (n) Z, p is a prime, f(x) is a polynomial of degree m irreducible over K, and (f(x)) is a principal ideal generated by the polynomial f(x) in the ring K[x]. Properties of the lattice of subrings of a Galois ring depend on values of numbers n and m. A subring lattice L of GR(p (n) , m) has the simplest structure for m = 1 (L is a chain) and for n = 1 (L is distributive). It turned out that only in these cases there are examples of projections of Galois ring onto rings that are not Galois rings. We prove the following result (Thm. 4). Let R = GR(p (n) , q (m) ), where n > 1 and m > 1. Then R (phi) a parts per thousand... R

On categorical equivalence of finite p-rings

“…Thus, F (I) ⊆ F (J). Lemma 3.3 allows us to apply results of Korobkov [11,12,13] about rngs with isomorphic subrng lattices. In particular, in the next proposition we determine prings that are categorically equivalent to a Galois p-ring.…”

“…Since every subrng of R is an ideal of some subring of R, it turns out that categorically equivalent rings have isomorphic subrng lattices. This is important because it allows us to apply the methods and results of Korobkov [11,12,13] who has extensively studied rngs with isomorphic subrng lattices.…”