2019/03/26 by Sharma, Pramod K
#13A15 #16D25 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.11067
Let A ⊂ B be rings. An ideal J ⊂ B is called power stable in A if Jn ∩ A = (J∩ A)n for all n≥ 1. Further, J is called ultimately power stable in A if Jn ∩ A = (J∩ A)n for all n large i.e., n ≫ 0. In this note, our focus is to study these concepts for pair of rings R ⊂ R[X] where R is an integral domain. Some of the results we prove are: A maximal ideal m in R[X] is power stable in R if and only if \wpt is \wp-primary for all t ≥ 1 for the prime ideal \wp = m∩ R. We use this to prove that for a Hilbert domain R, any radical ideal in R[X] which is a finite intersection of G-ideals is power stable in R. Further, we prove that if R is a Noetherian integral domain of dimension 1 then any radical ideal in R[X] is power stable in R, and if every ideal in R[X] is power stable in R then R is a field. We also show that if A ⊂ B are Noetherian rings, and I is an ideal in B which is ultimately power stable in A, then if I ∩ A = J is a radical ideal generated by a regular A-sequence, it is power stable. Finally, we give a relationship in power stability and ultimate power stability using the concept of reduction of an ideal (Theorem 3.22).