Centrally essential rings which are not necessarily unital or associative
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Viktor T. Markov
Abstract
Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.
Note: Originally published in Diskretnaya Matematika (2018) 30, N°4, 42–47 (in Russian).
Funding V. T. Markov was supported by the Russian Foundation for Basic Research, project 17-01-00895-A. A. A. Tuganbaev was supported by Russian Scientific Foundation, project 16-11-10013.
Acknowledgment
The authors are grateful to I.B. Kozhukhov for useful comments.
References
[1] Lam T.Y., A First Course in Noncommutative Rings, Springer, 2001.10.1007/978-1-4419-8616-0Suche in Google Scholar
[2] Markov V.T., Tuganbaev A.A., "Centrally essential rings", Discrete Math. Appl., 2019:3, 189–194.10.1515/dma-2019-0017Suche in Google Scholar
[3] Tuganbaev A.A., Ring Theory. Arithmetical Modules and Rings, MCCME, Moscow, 2009 (In Russian).Suche in Google Scholar
[4] Zhevlakov K.A., Slin'ko A.M., Shestakov I.P., Shirshov A.I., Rings that are nearly associative, Academic Press, New York-London, 1982, xi+371 pp.Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Centrally essential rings which are not necessarily unital or associative
- On the asymptotics of degree structure of configuration graphs with bounded number of edges
- Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme
- On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates
- Limit Poisson law for the distribution of the number of components in generalized allocation scheme
- Finite algebras of Bernoulli distributions
Artikel in diesem Heft
- Frontmatter
- Centrally essential rings which are not necessarily unital or associative
- On the asymptotics of degree structure of configuration graphs with bounded number of edges
- Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme
- On the complexity of bounded-depth circuits and formulas over the basis of fan-in gates
- Limit Poisson law for the distribution of the number of components in generalized allocation scheme
- Finite algebras of Bernoulli distributions
