DOI QR코드

DOI QR Code

ON 𝜙-n-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Mostafanasab, Hojjat ;
  • Darani, Ahmad Yousefian
  • Received : 2015.03.17
  • Published : 2016.05.01

Abstract

All rings are commutative with $1{\neq}0$ and n is a positive integer. Let ${\phi}:{\Im}(R){\rightarrow}{\Im}(R){\cup}\{{\emptyset}\}$ be a function where ${\Im}(R)$ denotes the set of all ideals of R. We say that a proper ideal I of R is ${\phi}$-n-absorbing primary if whenever $a_1,a_2,{\cdots},a_{n+1}{\in}R$ and $a_1,a_2,{\cdots},a_{n+1}{\in}I{\backslash}{\phi}(I)$, either $a_1,a_2,{\cdots},a_n{\in}I$ or the product of $a_{n+1}$ with (n-1) of $a_1,{\cdots},a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of ${\phi}$-n-absorbing primary ideals.

Keywords

n-absorbing ideals;n-absorbing primary ideals;${\phi}$-n-absorbing primary ideals

References

  1. D. D. Anderson and M. Batanieh, Generalizations of prime ideals, Comm. Algebra 36 (2008), no. 2, 686-696. https://doi.org/10.1080/00927870701724177
  2. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), no. 4, 831-840.
  3. D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39 (2011), no. 5, 1646-1672. https://doi.org/10.1080/00927871003738998
  4. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 417-429. https://doi.org/10.1017/S0004972700039344
  5. A. Badawi and E. Houston, Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains, Comm. Algebra 30 (2002), no. 4, 1591-1606. https://doi.org/10.1081/AGB-120013202
  6. A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51 (2014), no. 4, 1163-1173. https://doi.org/10.4134/BKMS.2014.51.4.1163
  7. A. Badawi, U. Tekir, and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc. 52 (2015), no. 1, 97-111. https://doi.org/10.4134/JKMS.2015.52.1.097
  8. A. Badawi and A. Yousefian Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39 (2013), no. 2, 441-452.
  9. S. M. Bhatwadekar and P. K. Sharma, Unique factorization and birth of almost primes. Comm. Algebra 33 (2005), no. 1, 43-49. https://doi.org/10.1081/AGB-200034161
  10. S. Ebrahimi Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J. 12 (2005), no. 3, 423-429.
  11. M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra 40 (2012), no. 4, 1268-1279. https://doi.org/10.1080/00927872.2010.550794
  12. D. J. Fieldhouse, Purity and Flatness, McGill University, Montreal, Canada, 1967.
  13. R. Gilmer, Multiplicative ideal theory, Queens Papers Pure Appl. Math. 90, Queens University, Kingston, 1992.
  14. J. Hukaba, Commutative rings with zero divisors, Marcel Dekker, Inc., New York, 1988.
  15. H. Mostafanasab, F. Soheilnia, and A. Yousefian Darani, On weakly n-absorbing ideals of commutative rings, submitted.
  16. H. Mostafanasab, E. Yetkin, U. Tekir, and A. Yousefian Darani, On 2-absorbing primary submodules of modules over commutative rings, An. St. Univ. Ovidius Constanta, (in press).
  17. H. Mostafanasab and A. Yousefian Darani, 2-irreducible and strongly 2-irreducible ideals of commutative rings, Miskolc Math. Notes, (in press).
  18. P. Nasehpour, On the Anderson-Badawi ${\omega}_{R[X]}$(I[X]) = ${\omega}_R$(I) conjecture, arXiv:1401.0459, (2014).
  19. D. G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Cambridge Phil. Soc. 55 (1959), 282-288. https://doi.org/10.1017/S030500410003406X
  20. J. Ohm and D. E. Rush, Content modules and algebras, Math. Scand. 31 (1972), 49-68. https://doi.org/10.7146/math.scand.a-11411
  21. P. A. J. Pena and J. E. Vielma, Isolated points and redundancy, Comment. Math. Univ. Carolin. 52 (2011), no. 1, 145-152.
  22. P. Quartararo and H. S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52 (1975), 91-96. https://doi.org/10.1090/S0002-9939-1975-0382249-5
  23. D. E. Rush, Content algebras, Canad. Math. Bull. 21 (1978), no. 3, 329-334. https://doi.org/10.4153/CMB-1978-057-8
  24. R. Y. Sharp, Steps in Commutative Algebra, Second edition, Cambridge University Press, Cambridge, 2000.
  25. A. Yousefian Darani, Generalizations of primary ideals in commutative rings, Novi Sad J. Math. 42 (2012), no. 1, 27-35.

Cited by

  1. Weakly Classical Prime Submodules vol.56, pp.4, 2016, https://doi.org/10.5666/KMJ.2016.56.4.1085