ON THE INTEGRAL CLOSURES OF IDEALS

• Journal title : Honam Mathematical Journal
• Volume 29, Issue 4,  2007, pp.653-666
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2007.29.4.653
Title & Authors
ON THE INTEGRAL CLOSURES OF IDEALS
Ansari-Toroghy, H.; Dorostkar, F.;

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 $\small{Ass_R}$(M), we obtain some results concerning $\small{I^{*(M)}}$ and prove that the sequence of sets $\small{Ass_R(R/(I^n)^{*(M)}}$$\small{)}$, $\small{n\;\in\;N}$, is increasing and ultimately constant. (Here $\small{(I^n)^{*(M)}}$ denotes the integral closure of $\small{I^n}$ relative to M.)
Keywords
reduction;integral dependence;integral closure;associated primes;
Language
English
Cited by
1.
TIGHT CLOSURE OF IDEALS RELATIVE TO MODULES,;;

호남수학학술지, 2010. vol.32. 4, pp.675-687
1.
TIGHT CLOSURE OF IDEALS RELATIVE TO MODULES, Honam Mathematical Journal, 2010, 32, 4, 675
2.
THE TIGHT INTEGRAL CLOSURE OF A SET OF IDEALS RELATIVE TO MODULES, Honam Mathematical Journal, 2016, 38, 2, 231
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.

2.

3.
Z. Elbast and P. Smith, Multiplication modules, Commun. in Algebra, (4) 16 (1988), 755-779.

4.
E. Matlis, Injective modules over Noetherian ring, Pacific J. Math., 8 (1958), 511-528.

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.

7.
D.G. Northcott and D. Rees, Reduction of ideals in local ring, Proc. Cambridge Philos. Soc., 50 (1954), 145-158.

8.
L.J. Ratliff, On asymptotic prime divisors, Pacific J. Math. 111 (1984), 395-413.

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.

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.

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.