△-CLOSURES OF IDEALS WITH RESPECT TO MODULES

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 1,  2008, pp.101-113
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.1.101
Title & Authors
△-CLOSURES OF IDEALS WITH RESPECT TO MODULES
Ansari-Toroghy, H.; Dorostkar, F.;

Abstract
Let M be an arbitrary module over a commutative Noetherian ring R and let $\small{{\triangle}}$ be a multiplicatively closed set of non-zero ideals of R. In this paper, we will introduce the dual notion of $\small{{\triangle}}$-closure and $\small{{\triangle}}$-dependence of an ideal with respect to M and obtain some related results.
Keywords
$\small{{\triangle}}$-reduction;$\small{{\triangle}}$-dependence;$\small{{\triangle}}$-closure;associated primes;attached primes;and asymptotic behavior;
Language
English
Cited by
References
1.
H. Ansari-Toroghy and R.Y. Sharp, Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings, Proc. Edinburgh Math. Soc. (2) 34 (1991), 155-160.

2.
H. Ansari-Toroghy and R.Y. Sharp, Integral closure of ideals relative to injective modules over commutative Noetherian rings, Quart. J. Math. Oxford, (2) 42 (1991), 393-402.

3.
L. Melkersson, P. Schenzel, Asymptotic attached prime ideals related to injective modules, Comm. Algebra (2) 20 (1992), 583-590.

4.
R. Naghipour and M. Sedghi, \${\Delta}-Reductions\$ and \${\Delta}-closures\$ of ideals with respect to an Artinian module, Communications in Algebra, 34 (2006), 763-777.

5.
L.J. Ratliff, Jr., \${\Delta}-closures\$ of ideals and rings, Trans. Amer. Math. Soc., 313 (1989), 221-247.

6.
L.J. Ratliff, Jr. and D. E. Rush, \${\Delta}-Reductions\$ of modules, Communications in Algebra, (8) 21 (1993), 2667-2685.

7.
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.