On σ-primeness and σ-semiprimeness in rings with involution

Year 2024, Volume: 13 Issue: 3, 262 - 270, 31.12.2024

Didem Yeşil , Didem Karalarlıoğlu Camcı , Barış Albayrak

https://doi.org/10.54187/jnrs.1581184

Abstract

This study introduces the structure of $\left\vert
\mathcal{S}_{R}^{\sigma}\right\vert $-$\sigma$-semiprime ring, $\left\vert \mathcal{S}_{R}^{\sigma}\right\vert $-$\sigma$-prime ring, and source of $\sigma$-primeness , which have not been previously explored, and presents new results. We define a subset $$
P_{R_{\sigma}} = \bigcap_{a\in R} S_{R_{\sigma}}^{a}
$$ of ring $R$ as $S_{R}^{\sigma}=\left\{ \left. a\in R\right\vert aRa=aR{\sigma}(a)=(0)\right\} $, where $\sigma$ is an involution is referred to as the source of $\sigma$-primeness of $R$. Additionally, we have established some relationships between the prime radical $\beta(R)$ and $\mathcal{S}_{R}^{\sigma}$.

Keywords

Involution, σ-Prime ring, σ-Semiprime ring, Source of primeness, Source of semiprimeness

Project Number

FBA-2019-2812

Thanks

The research has been supported by Çanakkale Onsekiz Mart University The Scientific Research Coordination Unit, Grant number: FBA-2019-2812.

References

• N. H. McCoy, The theory of rings, The Macmillan & Co LTD, New York, 1964.
• I. N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976.
• L. Oukhtite, L. Taoufiq, Some properties of derivations on rings with involution, International Journal of Modern Mathematical Sciences 4 (3) (2009) 309–315.
• M. Ashraf, S. Ali, On left multipliers and commutativity of prime rings, Demonstratio Mathematica 41 (4) (2008) 764–771.
• H. E. Bell, W. S. Martindale III, Centralizing mappings of semiprime rings, Canadian Mathematical Bulletin 30 (1) (1987) 92–101.
• L. Moln´ar, On centralizers of an H-algebra, Publicationes Mathematicae Debrecen 46 (1-2) (1995) 89–95.
• L. Oukhtite, On Jordan ideals and derivations in rings with involution, Commentationes Mathematicae Universitatis Carolinae 51 (3) (2010) 389–395.
• L. Oukhtite, Left multipliers and Lie ideals in rings with involution, International Journal of Open Problems in Computer Science and Mathematics 3 (3) (2010) 267–277.
• L. Oukhtite, Posner’s second theorem for Jordan ideals in ring with involution, Expositiones Mathematica 29 (4) (2011) 415–419.
• E. C. Posner, Derivations in prime rings, Proceedings of the American Mathematical Society 8 (6) (1957) 1093–1100.
• J. Mayne, Centralizing automorphisms of prime rings, Canadian Mathematical Bulletin 19 (1) (1976) 113–115.
• N. Aydın, Ç. Demir, D. Karalarlıo˘glu Camcı, The source of semiprimeness of rings, Communications of the Korean Mathematical Society 33 (4) (2018) 1083–1096.
• D. Yeşil, D. Karalarlıoğlu Camcı, The source of primeness of rings, Journal of New Theory (41) (2022) 100–104.
• L. Oukhtite, S. Salhi, Centralizing automorphisms and Jordan left derivations on σ-prime rings, Advances in Algebra 1 (1) (2008) 19–26.
• L. Oukhtite, S. Salhi, On commutativity of σ-prime rings, Glasnik Matematicki Series III 41 (61) (2006) 57–64.
• N. U. Rehman, R. M. Al-Omary, A. Z. Ansari, On Lie ideals of ∗-prime rings with generalized derivations, Boletin de la Sociedad Matematica Mexicana 21 (2015) 19–26.
• D. Karalarlıoğlu Camcı, Source of semiprimeness and multiplicative (generalized) derivations in rings, Doctoral Dissertation Çanakkale Onsekiz Mart University (2017) Çanakkale.
• D. Karalarlıoğlu Camcı, D. Yeşil, B. Albayrak, Source of semiprimeness of ∗-prime rings, Journal of Amasya University the Institute of Sciences and Technology 5 (1) (2024) 43–48.