Source of semiprimeness of $\ast$-prime rings

Year 2024, Volume: 5 Issue: 1, 43 - 48, 29.10.2024

Didem Karalarlıoğlu Camcı , Didem Yeşil , Barış Albayrak

https://doi.org/10.54559/jauist.1552047

Abstract

This study constructs a structure $S_{R}^{\ast}$ that had never been studied before and obtained new results by defining a subset $S_{R}^{\ast}$ of $R$ as
$S_{R}^{\ast}=\left\{ \left. a\in R\right\vert aRa=aRa^{\ast}=(0)\right\} $ where $\ast$ is an involution and it is called as the source of $\ast$-semiprimeness of $R$. Moreover, it investigates some properties of the subset $S_{R}^{\ast}$ in any ring $R$. Additionally, the relation between the prime radical, which provides the opportunity to work on prime rings, has been studied in many ways, and the set $\mathcal{S}_{R}^{\sigma}$, the motivation of this study, is provided. Furthermore, it is proved that $\mathcal{S}_{R}^{\sigma}=\{0\}$ in the case where the ring $R$ is a reduced ($\sigma$-semiprime) ring and $f(\mathcal{S}_{R}^{\sigma})=\mathcal{S}_{f(R)}^{\sigma}$ under certain conditions for a ring homomorphism $f$. Besides, it is presented that for the idempotent element $e$, the inclusion $e\mathcal{S}_{R}^{\sigma}e\subseteq\mathcal{S}_{eRe}^{\sigma}$ is provided, and for the right ideal (ideal) $I$ of the ring $R$, $\mathcal{S}_{R}^{\sigma}(I)$ is a left semigroup ideal (semigroup ideal) of the multiplicative semigroup $R$. In addition, it is analyzed that the set $\mathcal{S}_{R}^{\sigma}$ is contained by the intersection of all semiprime ideals of the ring $R$.

Keywords

Involution, ∗-prime ring, ∗-semiprime ring, source of semiprimeness

Supporting Institution

Çanakkale Onsekiz Mart Üniversitesi

Project Number

FBA-2019-2812

References

• I. N. Herstein, Rings with involution, Chicago lectures in mathematics, The University of Chicago Press, Chicago, 1976.
• 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.
• N. H. McCoy, The theory of rings, The Macmillan & Co LTD., New York, 1964.
• M. Ashraf, S. Ali, On left multipliers and commutativity of prime rings, Demonstratio Mathematica 4 (41) (2008) 764–771.
• H. E. Bell, W. S. Martindale III, Centralizing mappings of semiprime rings, Canadian Mathematical Bulletin 30 (1) (1987) 92–101.
• L. Molna´r, On centralizers of an H*-algebra, Publicationes Mathematicae Debrecen 46 (1-2) (1995) 89–95.
• E. C. Posner, Derivations in prime rings, Proceedings of the American Mathematical Society 8 (1957) 1093–1100.
• J. Mayne, Centralizing automorphisms of prime rings, Canadian Mathematical Bulletin 19 (1976) 113–115.
• N. Aydın, Ç. Demir, D. Karalarlıoğlu Camcı, The source of semiprimeness of rings, Communications of the Korean Mathematical Society 33 (4) (2018) 1083–1096.
• D. Karalarlıoğlu Camcı, Source of semiprimeness and multiplicative (generalized) derivations in rings, Doctoral Dissertation, Çanakkale Onsekiz Mart University (2017) Çanakkale.