Strong Persistence Index and Fluctuations in Colon Powers of Monomial Ideals

Let I be an ideal in a commutative Noetherian ring R. We say that a positive integer ℓ0 is the strong persistence index of I if ℓ0 is the smallest integer such that (Iℓ+1:RI)=Iℓ for all ℓ≥ℓ0. The first aim of this paper is to study this notion for monomial ideals. We also introduce the notion of fluctuation in colon powers if there exist positive integers a<b<c such that at least one of the following cases occurs: (i) (Ia:I)=Ia−1,(Ib:I)≠Ib−1,but(Ic:I)=Ic−1. (ii) (Ia:I)≠Ia−1,(Ib:I)=Ib−1,but(Ic:I)≠Ic−1. The second purpose of this work is to study this phenomenon for monomial ideals.