Geodesic
Lattice definability of certain matrix rings
Sbornik. Mathematics, Tome 208 (2017) no. 1, pp. 90-102

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Let $R=M_n(K)$ be the ring of square matrices of order $n\geqslant 2$ over the ring $K= \mathbb{Z}/p^k\mathbb{Z}$, where $p$ is a prime number, $k\in\mathbb{N}$. Let $R'$ be an arbitrary associative ring. It is proved that the subring lattices of the rings $R$ and $R'$ are isomorphic if and only if the rings $R$ and $R'$ are themselves isomorphic. In other words, the lattice definability of the matrix ring $M_n(K)$ in the class of all associative rings is proved. The lattice definability of a ring decomposable into a direct (ring) sum of matrix rings is also proved. The results obtained are important for the study of lattice isomorphisms of finite rings. Bibliography: 13 titles.
Keywords: lattice isomorphisms of associative rings, matrix rings, Galois rings. 
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     author = {S. S. Korobkov},
     title = {Lattice definability of certain matrix rings},
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     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_1_a4/}
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S. S. Korobkov. Lattice definability of certain matrix rings. Sbornik. Mathematics, Tome 208 (2017) no. 1, pp. 90-102. http://geodesic.mathdoc.fr/item/SM_2017_208_1_a4/