A Generalization of the Prime Radical of Rings

Yıl 2023, Cilt: 6 Sayı: 2 , 61 - 69 , 30.12.2023

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

https://doi.org/10.38061/idunas.1401075

https://izlik.org/JA45FS22MX

Öz

Let $R$ be a ring, $I$ be an ideal of $R$, and $\sqrt{I}$ be a prime radical of $I$. This study generalizes the prime radical of $\sqrt{I}$ where it denotes by $\sqrt[n+1]{I}$, for $n\in \mathbb{Z}^{+}$. This generalization is called $n$-prime
radical of ideal $I$. Moreover, this paper shows that $R$ is isomorphic to a subdirect sum of ring $H_{i}$ where $%
H_{i}$ are $n$-prime rings. Furthermore, two open problems are presented.

Anahtar Kelimeler

prime ring , prime ideal , prime radical , semiprime ideal

Kaynakça

• 1. Karalarlıoğlu Camcı, D. (2017). Source of semiprimeness and multiplicative (generalized) derivations in rings, Doctoral Thesis, Çanakkale Onsekiz Mart University, Çanakkale, Turkey.
• 2. Aydın, N., Demir, Ç., Karalarlıoğlu Camcı, D. (2018). The source of semiprimeness of rings, Communications of the Korean Mathematical Society, 33(4), 1083-1096.
• 3. Karalarlıoğlu Camcı, D., Yeşil, D., Mekera, R., Camcı, Ç. A Generalization of Source of Semiprimeness, Submitted.
• 4. Azumaya, G. (1948). On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem, Japanese Journal of Mathematics, 19, 525-547.
• 5. Baer, R. (1943). Radical ideals, American Journal of Mathematics, 65, 537-568.
• 6. Brown B., McCoy, N. H. (1947). Radicals and subdirect sums, American Journal of Mathematics, 67, 46-58.
• 7. Jacobson, N. (1945). The radical and semi-simplicity for arbitrary rings, American Journal of Mathematics, 76, 300-320.
• 8. Köthe, G. (1930). Die Strukture der Ringe deren Restklassenring nach den Radikal vollstandigreduzibel ist, Mathematische Zeitschrift, 32, 161-186.
• 9. Levitzki, J. (1943). On the radical of a ring, Bulletin of the American Mathematical Society, 49, 462-466.
• 10. McCoy, N. H. (1949). Prime ideals in general rings, American Journal of Mathematics, 71, 833-833.
• 11. McCoy, N. H. (1964). The Theory of Rings. The Macmillan Co.
• 12. Harehdashti, J. B., Moghimi, H. F. (2017). A Generalization of the prime radical of ideals in commutative rings, Communication of the Korean Mathematical Society, 32 (3), 543–552.
• 13. Clark, W. E. (1968). Generalized Radical Rings, Canadian Journal of Mathematics , 20, 88 - 94.