On the asymptotic sequences over an ideal

Let I be an ideal of a Noetherian ring R. Ratliff has shown that very many known results concerning R-sequences and classical grade have a valid analogue for asymptotic sequences and asymptotic grade. As a generalization of the concepts of asymptotic sequences and asymptotic grade, we introduce the concepts of asymptotic I-sequences and asymptotic I-grade. It is shown that they play a role analogous to asymptotic sequences and asymptotic grade.