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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

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(p(n),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

Пусть $R=M_n(K)$ - кольцо квадратных матриц порядка $n\geqslant 2$ над кольцом $K= \mathbb{Z}/p^k\mathbb{Z}$, где $p$ - простое число, $k\in\mathbb{N}$. Пусть $R'$ - произвольное ассоциативное кольцо. Доказано, что решетки подколец колец $R$ и $R'$ изоморфны тогда и только тогда, когда изоморфны сами кольца $R$ и $R'$. Иными словами, доказана решеточная определяемость кольца матриц $M_n(K)$ в классе всех ассоциативных колец. Доказана также решеточная определяемость кольца, разложимого в прямую (кольцевую) сумму матричных колец. Полученные результаты важны для изучения решеточных изоморфизмов конечных колец. Библиография: 13 названий.

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