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ON THE INTEGRAL CLOSURES OF IDEALS

Ansari-Toroghy, H.;Dorostkar, F.

  • Received : 2007.04.11
  • Accepted : 2007.11.18
  • Published : 2007.12.25

Abstract

Let R be a commutative Noetherian ring (with a nonzero identity) and let M be an R-module. Further let I be an ideal of R. In this paper, by putting a suitable condition on $Ass_R$(M), we obtain some results concerning $I^{*(M)}$ and prove that the sequence of sets $Ass_R(R/(I^n)^{*(M)})$, $n\;\in\;N$, is increasing and ultimately constant. (Here $(I^n)^{*(M)}$ denotes the integral closure of $I^n$ relative to M.)

Keywords

reduction;integral dependence;integral closure;associated primes

References

  1. H. Ansari-Toroghy and R.Y. Sharp, Integral closure of ideals relative to injective modules over commutative Noetherian rings, Quart. J. Math., (2) 42 (1991), 393-402. https://doi.org/10.1093/qmath/42.1.393
  2. N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading Mass. 1972.
  3. Z. Elbast and P. Smith, Multiplication modules, Commun. in Algebra, (4) 16 (1988), 755-779. https://doi.org/10.1080/00927878808823601
  4. E. Matlis, Injective modules over Noetherian ring, Pacific J. Math., 8 (1958), 511-528. https://doi.org/10.2140/pjm.1958.8.511
  5. S. McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 1023, Springer, Berlin, 1983.
  6. L. Melkerson and P. Schenzel, Asymptotic attached prime ideals related to injective modules, Comm. Algebra (2) 20 (1992), 583-590. https://doi.org/10.1080/00927879208824358
  7. D.G. Northcott and D. Rees, Reduction of ideals in local ring, Proc. Cambridge Philos. Soc., 50 (1954), 145-158. https://doi.org/10.1017/S0305004100029194
  8. L.J. Ratliff, On asymptotic prime divisors, Pacific J. Math. 111 (1984), 395-413. https://doi.org/10.2140/pjm.1984.111.395
  9. D. Rees and R.Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc.,(2) 18 (1978), 449-463. https://doi.org/10.1112/jlms/s2-18.3.449
  10. R.Y. Sharp and A.J. Taherizadeh, Reductions and integral closures of ideals relative to an Artinian module, J. London Math. Soc., (2) 37 (1988), 203-218. https://doi.org/10.1112/jlms/s2-37.2.203
  11. D.W. Sharpe and P. Vamos, Injective modules (Cambridge University Press, 1972).
  12. S. Yassemi, Coassociated primes, Commun. in Algebra, 23 (1995), 1473-1498. https://doi.org/10.1080/00927879508825288

Cited by

  1. TIGHT CLOSURE OF IDEALS RELATIVE TO MODULES vol.32, pp.4, 2010, https://doi.org/10.5831/HMJ.2010.32.4.675
  2. THE TIGHT INTEGRAL CLOSURE OF A SET OF IDEALS RELATIVE TO MODULES vol.38, pp.2, 2016, https://doi.org/10.5831/HMJ.2016.38.2.231