Kishor Shah

The ratliff-rush ideals in a noetherian ring

Ratliff and Rush show in particular that Ĩ is the largest ideal for which, for sufficiently large positive integers n, (Ĩ) = I and hence that ̃̃ I = Ĩ. We call regular ideals I for which I = Ĩ Ratliff–Rush ideals, and we call Ĩ the Ratliff–Rush ideal associated with I. It is easy to see that an element a of I : I is integral over I, in the sense that there is an equation of the form a + b1a k−1 + . . . + bk = 0, where bi ∈ I for i = 1, . . . , k. Therefore, the ideal Ĩ is always between I and the integral closure I ′ of I, and hence integrally closed ideals are Ratliff–Rush ideals. Ratliff and Rush observe [RR, (2.3.4)] that the powers of an invertible ideal are Ratliff–Rush ideals, so any principal ideal generated by a nonzerodivisor is a Ratliff–Rush ideal. They also prove the interesting fact that for any regular ideal I of R, there is a positive integer n such that for all k ≥ n, Ĩk = I [RR, (2.3.2)], i.e., all sufficiently high powers of a regular ideal are Ratliff–Rush. A regular ideal I is always a reduction of its associated Ratliff–Rush ideal Ĩ, in the sense that I(Ĩ) = (Ĩ) for some positive integer n. For the basic facts on reductions and reduction numbers of ideals, we refer the reader to [NR], [H1], and [H2]. In particular, if there is an element a of an ideal I for which aI = I then aR is called a principal reduction of I and the smallest n for which this equation holds is called the reduction number of I. We will call a regular ideal I stable iff it has a principal reduction with reduction number at most one, i.e., iff there is an element a of I for which

On the irreducible components of an ideal

Let I be an M -primary ideal in a local ring (R, M) and let irr(I) denote the set of irreducible components of I, where an ideal q is an irreducible component of I if q occurs as a factor in some decomposition of I as an irredundant intersection of irreducible ideals. We give several characterizations of the ideals in irr(I) and show that if J is an ideal between I and an irreducible component of I, then J is the intersection of ideals in irr(I). We also exhibit examples showing that there may exist irreducible ideals containing I that contain no ideal in irr(I). Also, we determine necessary and sufficient conditions that the pricipal ideal uR[u, tI] of the Rees ring R[u, tI] have a unique cover, and apply this to the study of the form ring of R with respect to I.

On the embedded primary components of ideals. IV

The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let I be an ideal in a local ring (R, M) that has M as an embedded prime divisor, and for a prime divisor P of I let ICp(I) be the set of irreducible components q of I that are P-primary (so there exists a decomposition of I as an irredundant finite intersection of irreducible ideals that has q as a factor). Then the main results show: (a) ICM(I) = U{ICM(Q); Q is a MEC of I} (Q is a MEC of I in case Q is maximal in the set of M-primary components of I); (b) if I = nf{qi; i = 1, ..., n} is an irredundant irreducible decomposition of I such that qi is M-primary if and only if i = 1, ..., k < n, then nf{qi; i = 1, ..., k} is an irredundant irreducible decomposition of a MEC of I, and, conversely, if Q is a MEC of I and if n{Qj;j = 1, ...,m} (resp., nf{qi; i = 1, ..., n}) is an irredundant irreducible decomposition of Q (resp., I) such that q1 . , qk are the M-primary ideals in {ql, ... , qn}, then m = k and (nf{qi; i =k + 1, .., n}) n (n{Q1; j = i, ,m}) is an irredundant irreducible decomposition of I; (c) ICM(I) = {q; q is maximal in the set of ideals that contain I and do not contain I: M}; (d) if Q is a MEC of I, then ICM(Q) = {q; Q C q E ICM(I)}; (e) if J is an ideal that lies between I and an ideal Q E ICM(I), then J = n{q; J C q E ICM(I)}; and, (f) there are no containment relations among the ideals in U{ICp(I); P is a prime divisor of I}.

Note on Cyclotomic Polynomials and Prime Ideals

Abstract Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ℤ-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ⊂ Q of A such that ht(Q/P) = 2.