A Generalization of Source of Semiprimeness

Year 2024, Issue: 49, 62 - 68, 31.12.2024

Didem Karalarlıoğlu Camcı , Didem Yeşil , Rasie Mekera , Çetin Camcı

https://doi.org/10.53570/jnt.1581076

Abstract

This paper characterizes the semigroup ideal $\mathcal{L}_{R}^{n}(I)$ of a ring $R$, where $I$ is an ideal of $R$, defined by $\mathcal{L}_{R}^{0}(I)=I$ and $\mathcal{L}_{R}^{n}(I)=\{a\in R \mid aRa\subseteq \mathcal{L}_{R}^{n-1}(I)\}$, for all $n\in \mathbb{Z}^+$, the set of all the positive integers. Moreover, it studies the basic properties of the set $\mathcal{L}_{R}^{n}(I)$ and defines $n$-prime ideals, $n$-semiprime ideals, $n$-prime rings, and $n$-semiprime rings. This study also investigates relationships between the sets $\mathcal{L}_{R}(I)$ and $\mathcal{L}_{R}^{n}(I)$ and exemplifies some of the related properties. It obtains the main results concerning prime rings and prime ideals by the properties of the set $\mathcal{L}_{R}^{n}(I)$.

Keywords

Source of semiprimeness, semiprime rings, semiprime ideals, prime rings, prime ideals

References

• G. Calugareanu, A new class of semiprime rings, Houston Journal of Mathematics 44 (1) (2018) 21-30.
• A. Hamed, A. Malek, S-prime ideals of a commutative ring, Contributions to Algebra and Geometry 61 (3) (2020) 533-542.
• A. Tarizadeh, M. Aghajani, On purely-prime ideals with applications, Communications in Algebra 49 (2) (2021) 824-835.
• D. D. Anderson, E. Smith, Weakly prime ideals, Houston Journal of Mathematics 29 (4) (2003) 831-840.
• K. K. Pathak, J. Goswami, S-Semiprime ideals and weakly S-semiprime ideals of rings, Palestine Journal of Mathematics 12 (4) (2023) 115-124.
• D. D. Anderson, M. Bataineh, Generalizations of prime ideals, Communications in Algebra 36 (2) (2008) 686-696.
• A. Abouhalaka, A note on weakly semiprime ideals and their relationship to prime radical in noncommutative rings, Journal of Mathematics 2024 (2024) Article ID 9142090 6 pages.
• A. Badawi, On weakly semiprime ideals of commutative rings, Contributions to Algebra and Geometry 57 (3) (2016) 589-597.
• C. Beddani, W. Messirdi, 2-prime ideals and their applications, Journal of Algebra and Its Applications 15 (03) (2016) 1650051 11 pages.
• S. Koc, Ü. Tekir, G. Ulucak, On strongly quasi primary ideals, Bulletin of the Korean Mathematical Society 56 (3) (2019) 729-743.
• D. D. Anderson, T. Dumitrescu, S-Noetherian rings, Communications in Algebra 30 (9) (2002) 4407-4416.
• A. Badawi, On 2-absorbing ideals of commutative rings, Bulletin of the Australian Mathematical Society 75 (3) (2007) 417-429.
• Z. Bilgin, M. L. Reyes, Ü. Tekir, On right S-Noetherian rings and S-Noetherian modules, Communications in Algebra 46 (2) (2018) 863-869.
• S. M. Bhatwadekar, P. K. Sharma, Unique factorization and birth of almost primes, Communications in Algebra 33 (1) (2005) 43-49.
• H. Ahmed, H. Sana, S-Noetherian rings of the forms $A[X]$ and $A[[X]]$, Communications in Algebra 43 (9) (2015) 3848-3856.
• K. Ajaykumar, B. S. Kiranagi, R. Rangarajan, Pullback of Lie algebra and Lie group bundles and their homotopy invariance, Journal of Algebra and Related Topics 8 (1) (2020) 15-26.
• R. Kumar, On characteristic ideal bundles of a Lie algebra bundle, Journal of Algebra and Related Topics 9 (2) (2021) 23-28.
• 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.
• N. H. McCoy, The theory of rings, Chelsea Publishing Company, New York, 1973.
• Y. S. Park, J. P. Kim, Prime and semiprime ideals in semigroups, Kyungpook Mathematical Journal 32 (3) (1992) 629-633.
• W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, 2nd Edition, Academic Press, San Diego, 2002.