Asymptotic Primes of Ratliff–Rush Closure of Ideals with Respect to Modules

Jafar Amjadi, Reza Naghipour
2008 Communications in Algebra  
This is a joint work with R. Naghipour. Let R be a commutative Noetherian ring, M a non-zero finitely generated R-module and I an ideal of R. The purpose of this paper is to develop the concept of Ratliff-Rush closure I (M ) of I with respect to M . It is shown that the sequence Ass R R/ I n (M ) , n = 1, 2, ..., of associated prime ideals is increasing and eventually stabilizes. This result extends Mirbagheri-Ratliffs main result in On the relevant transform and the relevant component of an
more » ... al, J. Algebra 111 (1987), 507-519. Furthermore, if R is local, then the operation I → I (M ) is a c * -operation on the set of ideals I of R, each ideal I has a minimal Ratliff-Rush reduction J with respect to M , and, if K is an ideal between J and I, then every minimal generating set for J extends to a minimal generating set of K.
doi:10.1080/00927870801941689 fatcat:xxsyquzak5drxdzidjlhej5cbi