Geodesic
Projections of finite nonnilpotent rings
Algebra i logika, Tome 58 (2019) no. 1, pp. 69-83.

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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 a projective image of a ring $R$. Whenever a lattice isomorphism $\varphi$ implies an isomorphism between $R$ and $R^\varphi$, we say theat 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.
Keywords: finite rings, matrix rings, subring lattices, lattice isomorphisms of rings. 
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S. S. Korobkov. Projections of finite nonnilpotent rings. Algebra i logika, Tome 58 (2019) no. 1, pp. 69-83. http://geodesic.mathdoc.fr/item/AL_2019_58_1_a4/

[1] S. S. Korobkov, “Reshetochnye izomorfizmy konechnykh kolets bez nilpotentnykh elementov”, Izv. Ural. gos. un-ta. Matem. i mekhan. Kompyuter, n., 22:4 (2002), 81–93

[2] S. S. Korobkov, “Proektirovaniya kolets Galua”, Algebra i logika, 54:1 (2015), 16–33 | Zbl

[3] S. S. Korobkov, “Proektirovaniya konechnykh odnoporozhdennykh kolets s edinitsei”, Algebra i logika, 55:2 (2016), 192–218 | Zbl

[4] S. S. Korobkov, “Reshetochnaya opredelyaemost nekotorykh matrichnykh kolets”, Matem. sb., 208:1 (2017), 97–110

[5] D. W. Barnes, “Lattice isomorphisms of associative algebras”, J. Aust. Math. Soc., 6:1 (1966), 106–121 | MR | Zbl

[6] A. V. Yagzhev, “Reshetochnaya opredelyaemost nekotorykh matrichnykh algebr”, Algebra i logika, 13:1 (1974), 104–116 | Zbl

[7] R. L. Kruse, D. T. Price, Nilpotent rings, Gordon and Breach Sci. Publ., New-York–London–Paris, 1969 | MR | Zbl

[8] S. S. Korobkov, “Periodicheskie koltsa s razlozhimymi v pryamoe proizvedenie reshetkami podkolets”, Issledovanie algebraicheskikh sistem po svoistvam ikh podsistem, ed. P. A. Freidman, Ural. gos. ped. un-t, Ekaterinburg, 1998, 48–59

[9] R. D. Schafer, “A remark on finite simple rings”, Am. Math. Mon., 60 (1953), 696–697 | MR | Zbl

[10] H. Dzhekobson, Stroenie kolets, IL, M., 1961

[11] V. R. McDonald, Finite rings with identity, Marcel Dekker, New York, 1974 | MR | Zbl

[12] S. S. Korobkov, “Proektirovaniya periodicheskikh nilkolets”, Izv. vuzov. Matem., 1980, no. 7, 30–38

[13] V. P. Elizarov, Konechnye koltsa. Osnovy teorii, Gelios, M., 2006