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 Algebra Logika, 2016, Volume 55, Number 2, Pages 192–218 (Mi al737)  Projections of finite one-generated rings with identity

S. S. Korobkov

Urals State Pedagogical University, ul. K. Libknekhta 9, Yekaterinburg, 620065 Russia

Abstract: 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 else a lattice isomorphism) of the ring $R$ onto the ring $R'$. A ring $R'$ is called the projective image of a ring $R$. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form $GR(p^n,m)$, where $m>1$ and $n>1$. We specify sufficient conditions for the projective image of a onegenerated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

Keywords: finite rings, one-generated rings, lattice isomorphisms of associative rings.

DOI: https://doi.org/10.17377/alglog.2016.55.203  Full text: PDF file (250 kB) References: PDF file   HTML file

English version:
Algebra and Logic, 2016, 55:2, 128–145 Bibliographic databases:  UDC: 512.552

Citation: S. S. Korobkov, “Projections of finite one-generated rings with identity”, Algebra Logika, 55:2 (2016), 192–218; Algebra and Logic, 55:2 (2016), 128–145 Citation in format AMSBIB
\Bibitem{Kor16} \by S.~S.~Korobkov \paper Projections of finite one-generated rings with identity \jour Algebra Logika \yr 2016 \vol 55 \issue 2 \pages 192--218 \mathnet{http://mi.mathnet.ru/al737} \crossref{https://doi.org/10.17377/alglog.2016.55.203} \transl \jour Algebra and Logic \yr 2016 \vol 55 \issue 2 \pages 128--145 \crossref{https://doi.org/10.1007/s10469-016-9383-8} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000382002800003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84981275755} 

• http://mi.mathnet.ru/eng/al737
• http://mi.mathnet.ru/eng/al/v55/i2/p192

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This publication is cited in the following articles:
1. S. S. Korobkov, “Projections of finite commutative rings with identity”, Algebra and Logic, 57:3 (2018), 186–200    2. S. S. Korobkov, “Projections of finite nonnilpotent rings”, Algebra and Logic, 58:1 (2019), 48–58    3. S. S. Korobkov, “Lattice isomorphisms of finite local rings”, Algebra and Logic, 59:1 (2020), 59–70    4. Kaarli K., Waldhauser T., “on Categorical Equivalence of Finite P-Rings”, Commun. Algebr., 49:4 (2021), 1437–1450     •  Number of views: This page: 128 Full text: 23 References: 22 First page: 7 Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021