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 Algebra Discrete Math., 2011, Volume 12, Issue 2, Pages 72–84 (Mi adm130)  RESEARCH ARTICLE

Generalized symmetric rings

G. Kafkasa, B. Ungora, S. Halıcıoglua, A. Harmancib

a Department of Mathematics, Ankara University, Turkey
b Department of Mathematics, Hacettepe University, Turkey

Abstract: In this paper, we introduce a class of rings which is a generalization of symmetric rings. Let $R$ be a ring with identity. A ring $R$ is called central symmetric if for any $a$, $b, c\in R$, $abc = 0$ implies $bac$ belongs to the center of $R$. Since every symmetric ring is central symmetric, we study sufficient conditions for central symmetric rings to be symmetric. We prove that some results of symmetric rings can be extended to central symmetric rings for this general settings. We show that every central reduced ring is central symmetric, every central symmetric ring is central reversible, central semmicommutative, 2-primal, abelian and so directly finite. It is proven that the polynomial ring $R[x]$ is central symmetric if and only if the Laurent polynomial ring $R[x, x^{-1}]$ is central symmetric. Among others, it is shown that for a right principally projective ring $R$, $R$ is central symmetric if and only if $R[x]/(x^n)$ is central Armendariz, where $n\geq 2$ is a natural number and $(x^n)$ is the ideal generated by $x^n$.

Keywords: symmetric rings, central reduced rings, central symmetric rings, central reversible rings, central semicommutative rings, central Armendariz rings, 2-primal rings. Full text: PDF file (265 kB) References: PDF file   HTML file

Bibliographic databases:  MSC: 13C99, 16D80, 16U80
Revised: 18.12.2011
Language:

Citation: G. Kafkas, B. Ungor, S. Hal{\i}c{\i}oglu, A. Harmanci, “Generalized symmetric rings”, Algebra Discrete Math., 12:2 (2011), 72–84 Citation in format AMSBIB
\Bibitem{KafUngHal11}
\by G.~Kafkas, B.~Ungor, S.~Hal{\i}c{\i}oglu, A.~Harmanci
\paper Generalized symmetric rings
\jour Algebra Discrete Math.
\yr 2011
\vol 12
\issue 2
\pages 72--84
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2952903}
\zmath{https://zbmath.org/?q=an:1259.16042}

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This publication is cited in the following articles:
1. Wei J., “Generalized Weakly Symmetric Rings”, J. Pure Appl. Algebr., 218:9 (2014), 1594–1603     2. Jung D.W., Kim N.K., Lee Ya., Ryu S.J., “on Properties Related To Reversible Rings”, Bull. Korean. Math. Soc., 52:1 (2015), 247–261     3. Bhattacharjee A., Chakraborty U.Sh., “On Some Generalizations of Reversible and Semicommutative Rings”, Int. Electron. J. Algebr., 22 (2017), 11–27     4. Wang Y., “Examples of Central Semicommutative Rings”, Kyungpook Math. J., 58:3 (2018), 427–432    5. Meng F., Wei J., “E-Symmetric Rings”, Commun. Contemp. Math., 20:3 (2018), 1750039     6. Meng F., Wei J., “Some Properties of E-Symmetric Rings”, Turk. J. Math., 42:5 (2018), 2389–2399     7. Kose H., Ungor B., Kurtulmaz Y., Harmanci A., “a Perspective on Amalgamated Rings Via Symmetricity”, Rings, Modules and Codes, Contemporary Mathematics, 727, eds. Leroy A., Lomp C., LopezPermouth S., Oggier F., Amer Mathematical Soc, 2019, 237–247    8. Bhattacharjee A., Chakraborty U.Sh., “Ring Endomorphisms Satisfying the Central Reversible Property”, Proc. Indian Acad. Sci.-Math. Sci., 130:1 (2020), 12     9. T. Subedi, D. Roy, “On a common generalization of symmetric rings and quasi duo rings”, Algebra Discrete Math., 29:2 (2020), 249–258  10. Gadelseed B., Wei J., Yao H., “Weakly Local Commutativity For Rings With Unity”, Quaest. Math., 43:10 (2020), 1367–1384     • Number of views: This page: 210 Full text: 121 References: 33 First page: 1 Contact us: math-net2022_01 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2022