Associative rings are considered. By a lattice isomorphism (or projection) of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of a ring R onto the subring lattice L(Rφ) of a ring Rφ. Let Mn(GF(pk)) be the ring of all square matrices of order n over a finite field GF(pk), where n and k are natural numbers, p is a prime. A finite ring R with identity is called a semilocal (primary) ring if R/RadR ≅ Mn(GF(pk)). It is known that a finite ring R with identity is a semilocal ring iff R ≅ Mn(K) and K is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if φ is a projection of a ring R = Mn(K), where K is an arbitrary finite local ring, onto a ring Rφ, then Rφ = Mn(K′), in which case K′ is a local ring lattice-isomorphic to the ring K. We thus prove that the class of semilocal rings is lattice definable.
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Translated from Algebra i Logika, Vol. 61, No. 2, pp. 180-200, March-April, 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.203.
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Korobkov, S.S. Projections of Semilocal Rings. Algebra Logic 61, 125–138 (2022). https://doi.org/10.1007/s10469-022-09681-z
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DOI: https://doi.org/10.1007/s10469-022-09681-z