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 lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Whenever a lattice isomorphism φ implies an isomorphism between R and Rφ, we say that the ring R is determined by its subring lattice. The present paper is a continuation of previous research on lattice isomorphisms of finite rings. We give a complete description of projective images of prime and semiprime finite rings. One of the basic results is the theorem on lattice definability of a matrix ring over an arbitrary Galois ring. Projective images of finite rings decomposable into direct sums of matrix rings over Galois rings of different types are described.
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Translated from Algebra i Logika, Vol. 58, No. 1, pp. 69-83, January-February, 2019.
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Korobkov, S.S. Projections of Finite Nonnilpotent Rings. Algebra Logic 58, 48–58 (2019). https://doi.org/10.1007/s10469-019-09524-4
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DOI: https://doi.org/10.1007/s10469-019-09524-4