Note on Noetherian filtrations

On the grade and cograde of a Noetherian filtration

“…It follows from the definition of a filtration φ that = Rad(0(n)) for all n > 1, so it follows from (2.1.11) (and since = B,s.ά(φ i (m)) for ί = 1, -9 g and for all positive integers n and m) that Rad(Φ (1) (n)) = Rad(Φ fI) (l)) for all ne^. Also, if I and J are ideals with the same radical, then it follows from (2.3.1) (resp., (2.3.3)) that J*(7) = J*(J) (resp., 1(1) = J(J)).…”

“…, g, then <srf w {Φ) = srf w (φ\ ' * φ g ) = {P e Spec(Λ); P e Ass(J?/(Φ (1) (n)) J for some nonzero n 6 Jί g ) = Ass(i?/(Φ (1) (n)) J /or αZZ Zαr^β n e ^.…”

“…[u, tl] and 7 is a given ideal of i?.) They have played an important role in many research papers, and there are many results concerning them in the literature.In several recent papers a number of important theorems concerning ideals in a Noetherian ring have been extended to Noetherian filtrations (e.g., see [1,13,14,15,25,28,29]). And in [6,26] a number of results concerning the asymptotic prime divisors of an ideal are extended to finite collections of ideals.…”

“…P such that I(R P )* + 3 is PCRp)*-primary}, -2(0) = .2(0(1)), and Ά(Φ) = J(Φ (1) (1)). The members of <H(I) (resp., J2(φ), J(Φ)) are called the quintessential prime divisors of I (resp., φ, Φ).…”

“…) is a collection of finitely many filtrations on R and m, n e Jί g , then we use (Φ {m) (n)) w to denote the weak integral closure of the ideal Φ (m) (n) = Φ (1) (τnn) = Φ (mn) (l) with respect to the filtration…”