Principal ideal properties of R-sequences and asymptotic sequences

AbstractIt is well known that if I and J are ideals of a ring R (commutative with identity) and b is a regular nonunit in R, then: (a) bnI : bnR = I, for all n ≥ 1; (b) if bnJ ⊆ bnI for some n ≥ 1, t hen J ⊆ I; (c) bn(I ∩ J) = bnI ∩ bnJ for all n ≥ 1; and, (d) if R is Noetherian, then Ass(itR)/I ⊆ Ass(R/bnI) ⊆ Ass(R/I) ∪ Ass(R/bR) for all n ≥ 1, and the second inclusion is an equality when height(I) ≥ 1 and every prime divisor of bR has height one. The main result shows that (a)–(d) continue to hold when b is replaced by an ideal B that is generated by an R-sequence contained in I and J. Then an asymptotic sequence version of these results is given