Symmetricity of rings relative to the prime radical
DOI:
https://doi.org/10.5269/bspm.51713Abstract
In this paper, we introduce and study a strict generalization of symmetric rings. We call a ring $R$ \textit{`$P$-symmetric' } if for any $a,\, b,\, c\in R,\, abc=0$ implies $bac\in P(R)$, where $P(R)$ is the prime radical of $R$. It is shown that the class of $P$-symmetric rings lies between the class of central symmetric rings and generalized weakly symmetric rings. Relations are provided between $P$-symmetric rings and some other known classes of rings. From an arbitrary $P$-symmetric ring, we produce many families of $P$-symmetric rings.
References
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2. H. Chen, Rings Related to Stable Range Conditions, Series in Algebra 11, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. https://doi.org/10.1142/8006
3. Z. Liang, Y. Gang, On weakly reversible rings, Acta Math. Univ. Comenianae 76(2), 189-192, (2007).
4. G. Kafkas, B. Ungor, S. Halicioglu, A. Harmanci, Generalized symmetric rings, Algebra Discrete Math. 12(2), 72-84, (2011).
5. H. Kose, B. Ungor, Semicommutativity of the rings relative to prime radical, Comment. Math. Univ. Carolin. 56(4), 401-415, (2015). https://doi.org/10.14712/1213-7243.2015.140
6. N. H. McCoy, The Theory of Rings, Chelsea Publishing Company, New York, 1973.
7. J. Lambok, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14, 359-368, (1971). https://doi.org/10.4153/CMB-1971-065-1
8. L. Ouyang, H. Chen, On weak symmetric rings, Comm. Algebra 38(2), 697-713, (2010). https://doi.org/10.1080/00927870902828702
9. M. B. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. 73(A), 14-17, (1997). https://doi.org/10.3792/pjaa.73.14
10. J. Wei, Generalized weakly symmetric rings, J. Pure Appl. Algebra 218, 1594-1603, (2014). https://doi.org/10.1016/j.jpaa.2013.12.011
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Published
2022-12-23
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