Let R and R φ be associative rings with isomorphic subring lattices and φ be a lattice isomorphism (a projection) of the ring R onto the ring R φ. We call R φ the projective image of a ring R and call the ring R itself the projective preimage of a ring R φ. 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 φ ≅ R.
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Translated from Algebra i Logika, Vol. 54, No. 1, pp. 16-33, January-February, 2015.
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Korobkov, S.S. Projections of Galois Rings. Algebra Logic 54, 10–22 (2015). https://doi.org/10.1007/s10469-015-9318-9
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DOI: https://doi.org/10.1007/s10469-015-9318-9