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Year 2023, Volume: 6 Issue: 2, 61 - 69, 30.12.2023
https://doi.org/10.38061/idunas.1401075

Abstract

References

  • 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.

A Generalization of the Prime Radical of Rings

Year 2023, Volume: 6 Issue: 2, 61 - 69, 30.12.2023
https://doi.org/10.38061/idunas.1401075

Abstract

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.

References

  • 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.
There are 13 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Didem Karalarlıoğlu Camcı 0000-0002-8413-3753

Didem Yeşil 0000-0003-0666-9410

Rasie Mekera 0000-0002-0092-2991

Çetin Camcı 0000-0002-0122-559X

Publication Date December 30, 2023
Submission Date December 6, 2023
Acceptance Date December 18, 2023
Published in Issue Year 2023 Volume: 6 Issue: 2

Cite

APA Karalarlıoğlu Camcı, D., Yeşil, D., Mekera, R., Camcı, Ç. (2023). A Generalization of the Prime Radical of Rings. Natural and Applied Sciences Journal, 6(2), 61-69. https://doi.org/10.38061/idunas.1401075