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 Mat. Sb. (N.S.), 1985, Volume 128(170), Number 3(11), Pages 383–402 (Mi msb2166)

On the representation of finite rings by matrices over commutative rings

Yu. N. Mal'tsev

Abstract: The author constructs an infinite series of finite rings $B$, $B^{(m)}$, $m\geqslant2$, which are not embeddable in rings of matrices over commutative rings, and describes their bases of identities and critical rings of the varieties they generate. He shows that finite rings from the ring varieties $\operatorname{var}B$, $\operatorname{var}B^{(m)}$, $m\geqslant2$, $m=(p-1)t+1$, are either representable by matrices over commutative rings or generate the respective varieties. Under a supplementary restriction on a variety $\mathfrak M$ with exponent $p^k$ it is shown that every finite ring from $\mathfrak M$ is representable by matrices over a commutative ring if and only if $\mathfrak M$ does not contain any of the rings $B$, $B^{(m)}$, $m\geqslant2$.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:2, 379–402

Bibliographic databases:

UDC: 512
MSC: Primary 16A44; Secondary 16A42, 16A44

Citation: Yu. N. Mal'tsev, “On the representation of finite rings by matrices over commutative rings”, Mat. Sb. (N.S.), 128(170):3(11) (1985), 383–402; Math. USSR-Sb., 56:2 (1987), 379–402

Citation in format AMSBIB
\Bibitem{Mal85} \by Yu.~N.~Mal'tsev \paper On the representation of finite rings by matrices over commutative rings \jour Mat. Sb. (N.S.) \yr 1985 \vol 128(170) \issue 3(11) \pages 383--402 \mathnet{http://mi.mathnet.ru/msb2166} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=815271} \zmath{https://zbmath.org/?q=an:0607.16015|0595.16011} \transl \jour Math. USSR-Sb. \yr 1987 \vol 56 \issue 2 \pages 379--402 \crossref{https://doi.org/10.1070/SM1987v056n02ABEH003042} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. L. A. Bokut', “Embedding of rings”, Russian Math. Surveys, 42:4 (1987), 105–138
2. Bokut L., “Some New Results in the Combinatorial Theory of Rings and Groups”, Lect. Notes Math., 1352 (1988), 34–43
3. A.Z. Anan'in, “On Representability of a Finite Local Ring”, Journal of Algebra, 228:2 (2000), 417
4. A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126
5. A. Mekei, “On the representation of finite rings by matrices over commutative rings”, J. Math. Sci., 197:4 (2014), 548–557
6. A. Mekei, L. Oyuuntsetseg, “Representation of some finite rings by matrices over commutative rings”, Algebra and Logic, 53:4 (2014), 287–297
7. A. Mekei, “Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring”, J. Math. Sci., 213:2 (2016), 254–267
8. L. A. Bokut, “Early history of the theory of rings in Novosibirsk”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2017, no. 2, 5–23
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