The Weak Subintegral Closure of an Ideal

Marie A. Vitulli and John V. Leahy University of Oregon Eugene, OR 97403

Abstract

For an ideal I of a ring A, we introduce the notion of the weak subintegral closure of I in an extension B of A. This gives an ideal of the weak normalization of A in B and enables us to identify the homogeneous components of the weak normalization of the Rees ring A[It] in B[t]. We get a new closure operator on ideals of A by considering the weak subintegral closure in A.

1 Introduction

The integral closure of a ring and the corresponding notions of integral closure, reduction, and analytic spread of an ideal have long been used to study the geometry of complex hypersurfaces [16]. Similar notions were developed for modules by Rees [9] and applied to singularity theory by Gaffney [3].

In 1970 a notion of seminormalization for commutative rings was introduced by Traverso [17]. Shortly thereafter Hamann [4] showed that a commutative ring with identity is seminormal in its total quotient ring if and only if it contains every element of its total quotient ring whose square and cube are in the ring. About the same time the operation of weak normalization was defined by Andreotti and Norguet [2] for complex analytic spaces and by Andreotti and Bombieri [1] for abstract schemes. A complex analytic space is weakly normal if and only if every globally defined continuous function that is regular outside the singular locus is, in fact, globally regular. In [5] the authors proved that a complex algebraic variety is weakly normal (as a complex anaytic space) if and only if all its local rings are seminormal.

Swan [15] defined an extension A ⊆ B of commutative rings to be subintegral if B is integral over A and the induced map Spec(B) → Spec(A) is a bijection that induces isomorphisms on the residue field extensions. Yanagihara [18] defined an extension A ⊆ B of commutative rings to be weakly subintegral if B is integral over A and the induced map Spec(B) → Spec(A) is a bijection that makes the residue field extensions purely inseparable.

Roberts and Singh [14] defined an element b ∈ B to be subintegral over a Q-subalgebra A if and only if there exist elements c_{1}, …, c_{p} in B, and N ∈ N such that b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ A for all n ≥ N. They showed b is subintegral over A if and only if A ⊆ A[b] is a subintegral extension and that the set of all elements of B that are subintegral over A is the seminormalization of A in B. This was the first element-wise characterization of subintegrality that appeared in the literature. In a later paper [13] Reid, Roberts, and Singh eliminated the assumption that the base ring be a Q-algebra and showed that an element is quasisubintegral over a subring, i.e., satisfies the above condition, if and only if the element satisfies equations of integral dependence like those the current authors present in (2.1). In the same paper Reid, Roberts, and Singh also showed that the set of all elements B that are quasisubintegral over the subring A is the weak normalization of A in B. More recently Reid and Roberts [11] substituted the terminology “weakly subintegral element” for the older terminology “quasisubintegral element.” We adopt the more recent terminology for this paper.

In [6] the authors investigated the seminormality of reduced graded rings in preparation for studying the weak normalization of projective varieties and the procedure of blowing up a nonweakly normal variety along the nonnormal locus. This led naturally to the following question. Given an ideal in a reduced ring with finitely many minimal primes, what is the seminormalization of the Rees algebra associated with that ideal? Recently Oda and Yoshida [7] defined the (2,3)-closure I′ of an ideal I of a Noetherian domain A with finite normalization and showed that I′ is an ideal of the seminormalization of A. In a correction to the original paper [8], Yoshida and Oda showed that if (I′)^{n} = (I^{n})′ for all positive integers n, then the seminormalization of the Rees ring associated with I is the direct sum of the ideals (I^{n})′. We point out that the construction of Oda and Yoshida may be extended to define the (2,3)-closure of an ideal of a reduced ring with finitely many minimal primes.

In this paper we define an element b of an extension B of a ring A to be weakly subintegral over an ideal I of A if b satisfies a highly structured sequence of equations of integral dependence. After giving several equivalent characterizations of an element that is weakly subintegral over an ideal we define the weak subintegral closure of I in B. This gives an ideal of the weak normalization of A in B and enables us to identify the homogeneous components of the weak normalization of the Rees ring A[It] in B[t]. In addition, by taking the weak subintegral closure of an ideal I in A we get a new closure operator on ideals of A. We conclude by giving examples that distinguish the weak subintegral closure, the integral closure, and the radical of an ideal.

2 The Weak Subintegral Closure of an Ideal

Conventions. All rings are assumed to be commutative with identity. If a set S is properly contained in a set T we write S ⊂ T. We let N denote the set of positive integers and Z^{+} the set of nonnegative integers. We often abbreviate a polynomial expression f(x_{1}, …, x_{p}, y_{1}, …, y_{q}) by f(x, y) and a finitely generated R−algebra R[x_{1}, …, x_{p}, y_{1}, …, y_{q}] by R[x, y].

To aid the reader we review some notation and definitions that have appeared in the literature.

An extension of rings A ⊆ B is subintegral (respectively, weakly subintegral) if B is integral over A and the associated map Spec(B) → Spec(A) is a bijection that makes the extensions of residue class fields isomorphisms (respectively, purely inseparable extensions).

The seminormalization (respectively, weak normalization) of A in B is denoted by _{B}^{+}A (respectively, _{B}^{∗}A) and is the largest subintegral (respectively, weakly subintegral) extension of A in B.

Let A be a reduced ring and assume the total quotient ring K of A is a product of fields. Then _{K}^{+}A (respectively, _{K}^{∗}A) is denoted simply by^{+}A (respectively, ∗A) and is called the seminormalization (respectively, weak normalization) of A.

If b ∈ B and b^{2}, b^{3} ∈ A (respectively, b^{q}, qb ∈ A for some prime q), the extension A ⊆ A[b] is called elementary subintegral (respectively, elementary weakly subintegral). We recall that _{B}^{+}A (respectively, _{B}^{∗}A) is the filtered union of all subrings of B that can be obtained from A by a finite sequence of elementary subintegral (respectively, elementary subintegral or elementary weakly subintegral) extensions.

Let t be an indeterminate and let

\binom{t}{i} = \frac{t(t − 1) … (t − i + 1)}{i!} for i > 0 and \binom{t}{0} = 1.

This defines a polynomial of degree i with rational coefficients. Evaluating this polynomial at a nonnegative integer n ≥ i gives the usual binomial coefficient n choose i and evaluating it at a nonnegative integer n < i gives 0. For n > 0 we have

\binom{−n}{i} = (−1)^{i}\binom{n + i − 1}{i}.

Consequently, this polynomial takes integer values on the set of all integers.

Definition 2.1 Let A ⊆ B be an extension of rings. For an ideal I of A we say b ∈ B is weakly subintegral over I if there exists an integer p ∈ Z^{+} and elements a_{i} ∈ I^{i} (1 ≤ i ≤ 2p + 1) such that

b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i} = 0 (p + 1 ≤ n ≤ 2p + 1).

We let B^{∗}I denote the set of all elements of B that are weakly subintegral over I. We call B^{∗}I the weak subintegral closure of I in B. We define the weak subintegral closure of I to be the weak subintegral closure of I in A and denote this by ^{∗}I. We say I is weakly subintegrally closed in B if I is equal to B^{∗}I. We say I is weakly subintegrally closed if I is equal to ^{∗}I.

Roberts and Singh [14] defined what is means for an element to be subintegral over a subring A that contains a copy of Q and showed that the seminormalization of a reduced Q-algebra A is the set of all elements in the quotient field of A that are subintegral over A [14, (4.19)]. Their definition did not involve equations of integral dependence but rather a condition like the one that appears below in (2.9.i). In a later paper [13] Reid, Roberts, and Singh eliminated the assumption that the base ring contain a copy of the rationals and defined what it means for an element to be quasisubintegral over an subring. In this later paper it was shown that an element is quasisubintegral over a subring if and only if the element satisfies equations of integral dependence like those in (2.1) [13, (5.1) and remarks immediately thereafter] and that the set of all elements of B that are quasisubintegral over the subring A of B is the weak normalization of A in B [13, (6.11)]. So our notation is consistent with established notation. Notice that our definition differs from that given by Roberts and Singh in that we absorbed a factor a (−1)^{i} in the coefficient a_{i}. More recently Reid and Roberts [11] replaced the terminology “quasisubintegral element” by “weakly subintegral element.” We prefer the newer terminology and use it throughout this paper, even to describe old results.

Remark 2.2 It follows easily from (2.1) that if a is a regular element of a ring A whose total quotient ring K is a product of fields and A is weakly normal, then the ideal generated by a is weakly subintegrally closed in K. For suppose an element b ∈ K is weakly subintegral over I = (a). Since the weak normalization of A is equal to the weak subintegral closure of A in K, we must have b ∈ A. Choose a nonnegative integer p and elements d_{i} ∈ I^{i} (1 ≤ i ≤ 2p + 1) such that

b^{n} + ∑_{i=1}^{n} \binom{n}{i}d_{i}b^{n−i} = 0 (p + 1 ≤ n ≤ 2p + 1).

Write d_{i} = a^{i}c_{i}, with c_{i} ∈ A (1 ≤ i ≤ 2p + 1). Then,

(\frac{b}{a})^{n} + ∑_{i=1}^{n} \binom{n}{i}c_{i}(\frac{b}{a})^{n−i} = 0 for p + 1 ≤ n ≤ 2p + 1.

Thus \frac{b}{a} is weakly subintegral over A and hence is in A by [13, (6.11)]. Hence b ∈ aA, as asserted.

In order to show that B^{∗}I is an ideal of the weak normalization _{B}^{∗}A we need to establish several equivalent characterizations of elements that are weakly subintegral over an ideal. In turn this requires several technical lemmas. The lemmas which follow are based on results that appeared in various forms and versions in the papers Roberts, Singh, and Reid. Since the ideas of Roberts, Singh, and Reid evolved and became refined over a series of papers, we have chosen to aid the reader by including our own versions of their final arguments, suitably modified to fit the situation at hand.

Notation 2.3 Let d be a nonnegative integer. For each integer i with 0 ≤ i ≤ d let

π_{i}(t) = (−1)^{i}\binom{t + i − 1}{i}\binom{t + d}{d − i}.

Notice that each π_{i}(t) is a polynomial of degree d, π_{i}(−j) = δ_{ij} for 0 ≤ j ≤ d, where δ is the Kronecker delta, and π_{i}(n) ∈ Z for each n ∈ Z.

Lemma 2.4 Let d be a positive integer, A an integral domain containing a copy of Q, t an indeterminate, and F (t), G(t) ∈ A[t] with deg(F (t)G(t)) ≤ d. Then,

∑_{i=0}^{d} π_{i}(t)F (t + i)G(t − i) = F (0)G(2t).

Proof. (see [10, (1.1)]) Let X be another indeterminate and set

H(X) = ∑_{i=0}^{d} π_{i}(X)F (t + i)G(t − i) and K(X) = F (t − X)G(t + X).

Then, H(−j) = F (t + j)G(t − j) = K(−j) for 0 ≤ j ≤ d. As polynomials in X, both H and K have degree at most d, so we may deduce that H(X) = K(X). The assertion follows by substituting t for X.

Notation 2.5 Let p be a positive integer, x_{1}, …, x_{p}, z be indeterminates, set S′ = Z[x_{1}, …, x_{p}, z, z^{−1}], and S = Z[x_{1}, …, x_{p}, z]. Define a graded structure on S′ by declaring that weight(x_{i}) = i for 1 ≤ i ≤ p and weight(z) = 1. Then S is a graded subring of S′. Let γ_{n}(x_{1}, …, x_{p}, z) = γ_{n} = z^{n} + ∑_{i=1}^{p} \binom{n}{i}x_{i}z^{n−i} for n ≥ 0. Notice that γ_{n} is homogeneous of weight n.

Lemma 2.6 Let p be a positive integer and let γ_{n} be as in (2.5).

1. Define integers a_{i}(n) by

a_{i}(n) = \{ \begin{matrix}π_{i}(n) + π_{2p−i}(n), & if 0 ≤ i ≤ p − 1 \\ π_{p}(n), & if i = p \end{matrix}

where the polynomials π_{i}(t) are as in (2.3) with d = 2p. Then,

∑_{i=0}^{p} a_{i}(n)γ_{n+i}γ_{n+2p−i} = γ_{2n+2p}, for all n ≥ 0.

2. Define integers b_{i}(n) by b_{i}(n) = π_{i}(n) + π_{2p+1−i}(n) for 0 ≤ i ≤ p, where we are considering the polynomials π_{i}(t) defined in (2.3) with d = 2p + 1. Then,

∑_{i=0}^{p} b_{i}(n)γ_{n+i}γ_{n+2p+1−i} = γ_{2n+2p+1} for all n ≥ 0.

Proof. (see [10, (1.2)]) Let F (t) = 1 + ∑_{i=1}^{p} \binom{t}{i}x_{i}z^{−1}. Notice that we have F (n) = γ_{n}z^{−n} for all n ≥ 0. We now prove the first assertion. We first let G(t) = F (t + 2p). By (2.4) we have that

F (0)G(2t) = ∑_{i=0}^{2p} π_{i}(t)F (t + i)G(t − i)

so that

γ_{2n+2p}z^{−2n−2p} = F (0)G(2n) = ∑_{i=0}^{2p} π_{i}(n)F (n + i)G(n − i).

Observe that

F (n + i)G(n − i) = γ_{n+i}z^{−n−i}γ_{n+2p−i}z^{−n−2p+i}
= γ_{n+i}γ_{n+2p−i}z^{−2n−2p}
= F (n + (2p − i))G(n − (2p − i)).

Consequently,

∑_{i=0}^{2p} π_{i}(n)F (n + i)G(n − i) =

∑_{i=0}^{p−1}(π_{i}(n) + π_{2p−i}(n))aγ_{n+i}γ_{n+2p−i}z^{−2n−2p} + π_{p}(n)γ_{n+p}γ_{n+p}z^{−2n−2p},

and the assertion follows immediately.

To prove the second assertion set G(t) = F (t + 2p + 1) and proceed as in the proof of the first assertion.

Lemma 2.7 With notation and assumptions as above,

Z[{γ_{n}}_{n≥N}] = Z[{γ_{n}}_{N≤n≤2N+2p−1}],

for each positive integer N.

Proof. (see [13, (2.2)]) Set R′ = Z[{γ_{n}}_{n≥N}] and R = Z[{γ_{n}}_{N≤n≤2N+2p−1}]. Just suppose that R ⊂ R′. Let d ≥ N be the smallest integer with γ_{d} ∉ R. Then d > 2N + 2p − 1 so we can express d as d = 2n + 2p or d = 2n + 2p + 1 with n ≥ N. By (2.6) we have

γ_{d} ∈ Z[{γ_{i}|n ≤ i ≤ n + 2p + 1}] ⊆ R,

which is a contradiction.

Corollary 2.8 Let p ≥ 1, A ⊆ B an extension of rings, I an ideal of A, and c_{1}, …, c_{p}, b ∈ B. If γ_{n}(c_{1}, …, c_{p}, b) ∈ I^{n} for N ≤ n ≤ 2N + 2p − 1, then γ_{n}(c_{1}, …, c_{p}, b) ∈ I^{n} for all n ≥ N.

Proof. Just suppose γ_{n}(c, b) ∉ I^{n} for some n ≥ N. Let d ≥ N be the smallest integer with γ_{d}(c, b) ∉ I^{d}. Then d > 2N +2p−1 so we may write d = 2n+2p+ϵ, where ϵ = 0 or 1, and n ≥ N. Say d = 2n + 2p. By (2.7)

γ_{d} = γ_{2n+2p} = ∑_{i=0}^{p} a_{i}(n)γ_{n+i}γ_{n+2p−i}

so that

γ_{d}(c, b) = ∑_{i=0}^{p} a_{i}(n)γ_{n+i}(c, b)γ_{n+2p−i}(c, b).

Since we know that γ_{n+i}(c, b) ∈ I^{n+i} and also that γ_{n+2p−i} ∈ I^{n+2p−i} we have γ_{n+i}(c, b)γ_{n+2p−i}(c, b) ∈ I^{2n+2p} and hence γ_{d}(c, d) ∈ I^{2n+2p} = I^{d}, a contradiction. When d = 2n+2p+1 we argue in a similar mannerto reach a contradiction.

Let B be a ring and t an indeterminate. As in [13] we define multiplication in the formal series ring B[[t]] by declaring

t^{n} ∗ t^{m} = \binom{n + m}{m}t^{n+m},

and extending by linearity and completion. With this multiplication the coefficient of t^{n} in the product (∑_{n≥0} a_{n}t^{n})*(∑_{n≥0} b_{n}t^{n}) is ∑_{i=0}^{n} \binom{n}{i}a_{i}b_{n−i}. For an element b ∈ B we define Exp(b) by

Exp(b) = ∑_{n≥0} b^{n}t^{n}.

Observe that for elements b and c of B we have

Exp(b) ∗ Exp(c) = Exp(b + c);

in particular, Exp(b)∗Exp(−b) = 1.

We need to prove another technical lemma, which is motivated by a blend of arguments used in [13, proof of (2.1) and (2.3)]. After this lemma we will offer several equivalent conditions that characterize weakly subintegral elements.

Lemma 2.9 Let A ⊆ B be an extension of rings, I an ideal of A, and b ∈ B. Suppose there exist integers q ∈ Z^{+}, N ∈ N, indeterminates U, x_{1}, …, x_{q}, z, a polynomial G(U, x_{1}, …, x_{q}, z) ∈ Q[U, x_{1}, …, x_{q}, z^{−1}] and d_{1}, …, d_{q} ∈ B such that

G(0, x, z) = 1
z^{n}G(n, x, z) ∈ Z[x, z] ∀n ≥ 1; and
b^{n}G(n, d, b) ∈ I^{n} ∀n ≥ N

Then, there exists an integer p ∈ Z^{+} and elements c_{1}, …, c_{p} ∈ B such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ I^{n} (n ≥ 1).

Proof. Suppose such a G exists. Let τ(U) = 1 + ∑_{i=1}^{N−1}(−1)^{i}\binom{U}{i}. Notice that τ(n) = 0 for 1 ≤ n ≤ N − 1. Replacing G(U, x, z) by τ(U)G(U, x, z) we may and shall assume that b^{n}G(n, d, b) ∈ I^{n} ∀n ≥ 1.

By the first condition on G, we may write

G(U, x, z) = 1 + ∑_{i=1}^{p} \binom{U}{i}σ_{i},

where σ_{i} ∈ Q[x, z^{−1}] (i = 1, …, p). Determining the coefficients σ_{i} recursively by setting U = 1, 2, … and using the second and third conditions on G we see that σ_{i} = η_{i}z^{−i}, where η_{i} ∈ Z[x_{1}, …, x_{q}] (i = 1, …, p). Thus

G(U, x, z) = 1 + ∑_{i=1}^{p} \binom{U}{i}η_{i}z^{−i} and

z^{n}G(n, x, z) = z^{n} + ∑_{i=1}^{p} \binom{n}{i}η_{i}z^{n−i}.

Letting c_{i} = η_{i}(d_{1}, …, d_{q}, b) ∈ B (i = 1, …, p), we have

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} = b^{n}G(n, d, b) ∈ I^{n} for all n ≥ 1.

Theorem 2.10 Let I ⊆ A ⊆ B be as above and b ∈ B. An element b ∈ B is in the weak subintegral closure of I in B if any of the following equivalent conditions holds:

1. ∃p ∈ Z^{+} and a_{i} ∈ I^{i} (1 ≤ i ≤ 2p + 1) such that

b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i} = 0 (p + 1 ≤ n ≤ 2p + 1);

2. ∃p ∈ Z^{+} and a_{i} ∈ I^{i} (i ≥ 1) such that

b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i} = 0 (n > p);

3. ∃p ∈ Z^{+}, elements c_{1}, …, c_{p} in B, and N ∈ N such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ I^{n} (N ≤ n ≤ 2N + 2p − 1);

4. ∃p ∈ Z^{+}, elements c_{1}, …, c_{p} in B, and N ∈ N such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ I^{n} (n ≥ N);

5. ∃p ∈ Z^{+} and elements c_{1}, …, c_{p} in B such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ I^{n} (1 ≤ n ≤ 2p + 1);

6. ∃p ∈ Z^{+}, elements c_{1}, …, c_{p} in B such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} ∈ I^{n} (n ≥ 1);

7. ∃p, s ∈ Z^{+}, elements c_{1}, …, c_{p} in B, and N ∈ N, with N ≥ p + s, such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−s−i} ∈ I^{n} (n ≥ N);

8. ∃a_{i} ∈ I^{i} (i ≥ 1) such that

(1 + ∑_{n≥1} a_{n}t^{n}) ∗ Exp(b) ∈ B[t] and;

9. ∃q ∈ Z^{+}, N ∈ N, indeterminates U, x_{1}, …, x_{q}, z, a polynomial G(U, x_{1}, …, x_{q}, z) ∈ Q[U, x_{1}, …, x_{q}, z^{−1}] and d_{1}, …, d_{q} ∈ B such that

G(0, x, z) = 1
z^{n}G(n, x, z) ∈ Z[x, z] ∀n ≥ 1; and
b^{n}G(n, d, b) ∈ I^{n} ∀n ≥ N

Proof. Assume (8) holds. Since the coefficient of t^{n} in (1+∑_{n≥1} a_{n}t^{n})∗Exp(b) is b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i} and this coefficient is zero for n > p, condition (2) holds. Thus (8)⇒(2), and similarly (2)⇒(8).

Again assume (8) holds and let

∑_{n=0}^{p}(−1)^{n}c_{n}t^{n} = (∑_{n≥0} a_{n}t^{n}) ∗ Exp(b),

where a_{0} = c_{0} = 1. Then

(∑_{n=0}^{p}(−1)^{n}c_{n}t^{n}) ∗ Exp(−b) = ∑_{n≥0} a_{n}t^{n}.

Equating the coefficients of t^{n} in the latter expression and multiplying by (−1)^{n} gives (5). Thus (8)⇒(5).

Clearly (5)⇒(3) and (3)⇒(4) follows from (2.8).

(4)⇒(2): Suppose ∃c_{1}, …, c_{p} ∈ B and N ∈ N such that

b^{n} + ∑_{i=1}^{p} \binom{n}{i}c_{i}b^{n−i} =: (−1)^{n}a_{n} ∈ I^{n}, for all n ≥ N.

Then for each n ≥ N we have,

b^{n+i} + ∑_{j=1}^{p} \binom{n + i}{j}c_{j}b^{n+i−j} = (−1)^{n+i}a_{n+i} ∈ I^{n+i} (0 ≤ i ≤ p).

Multiplying by b^{p−i} we obtain

b^{n+p} + ∑_{j=1}^{p} \binom{n + i}{j}c_{j}b^{n+p−j} = (−1)^{n+i}a_{n+i}b^{p−i} (0 ≤ i ≤ p).

Let M denote the matrix whose (i, j) entry is \binom{n+i}{j} for 0 ≤ i, j ≤ p and let M_{i0} denote the submatrix obtained from M by deleting row i and column 0. Then det M = 1 (see discussion after (4.10) in [12]). Applying Cramer’s Rule and expanding the resulting determinant down column 0 we obtain

b^{n+p} = ∑_{i=0}^{p}(−1)^{i}(−1)^{n+i}a_{n+i}b^{p−i}detM_{i0}.

As noted in the discussion after (5.5) in [13], detM_{i0} = \binom{n+i−1}{i}\binom{n+p}{p−i} so

b^{n+p} − (−1)^{n} ∑_{i=0}^{p} \binom{n + p}{p − i}\binom{n + i − 1}{i}a_{n+i}b^{p−i} = 0.

Since this holds for each n ≥ N, condition (2) holds. Thus (4)⇒(2). Consequently, the statements numbered (2)-(5) and (8) are equivalent.

Clearly (4) implies (7). Statement (7) implies (4) from (2.9). Thus the statements numbered (2)-(5) and (7)-(8) are equivalent.

Obviously (2)⇒(1).

(1)⇒(5): Assume (1) holds. Define c_{n} = (−1)^{n} (b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i}) for 1 ≤ n ≤ p. Then

(∑_{n=0}^{p}(−1)^{n}c_{n}t^{n}) ∗ Exp(−b) = ∑_{n≥0}(−1)^{n} (∑_{i=0}^{p} \binom{n}{i}c_{i}b^{n−i}) t^{n}.

We also have

( ∑_{n=0}^{2p+1} a_{n}t^{n}) ∗ Exp(b) = ∑_{n=0}^{p}(−1)^{n}c_{n}t^{n} + O(t^{2p+2}).

Thus

(∑_{n=0}^{p}(−1)^{n}c_{n}t^{n}) ∗ Exp(−b) = ∑_{n=0}^{2p+1} a_{n}t^{n} + O(t^{2p+2}).

Consequently,

∑_{i=0}^{p} \binom{n}{i}c_{i}b^{n−i} = (−1)^{n}a_{n} ∈ I^{n} (1 ≤ n ≤ 2p + 1).

Hence (1) implies (5). Thus after deleting (6), statements (1)-(8) are equivalent.

From Lemma 2.9 we may deduce that (4)⇒(9) by considering G(U, x_{1}, …, x_{p}, z) = 1 + ∑_{i=1}^{p} \binom{U}{i}x_{i}z^{−i} and also that (9)⇒(6). It is obvious that (6)⇒(7). Thus (1)-(9) are equivalent.

We will now show that the weak subintegral closure of an ideal I in an extension ring B of A is an ideal of the weak normalization of A in B.

Proposition 2.11 _{B}^{∗}I is an ideal of _{B}^{∗}A. Furthermore, for any ideals I and J of A we have (_{B}^{∗}I)(_{B}^{∗}J) ⊆ _{B}^{∗}(IJ). Consequently, ⊕_{n≥0} _{B}^{∗}(I^{n})t^{n} is a graded subring of _{B}^{∗}A[t].

Proof. Say c and d are elements of _{B}^{∗}I. Set a_{0} = b_{0} = 1 and choose a_{n}, b_{n} ∈ I^{n} for each n ≥ 1 with

(∑_{n≥0} a_{n}t^{n}) ∗ Exp(c), (∑_{n≥0} b_{n}t^{n}) ∗ Exp(d) ∈ B[t].

Then

(∑_{n≥0} e_{n}t^{n}) ∗ Exp(c + d) ∈ B[t],

where e_{n} = ∑_{i=0}^{n} \binom{n}{i}a_{i}b_{n−i}. Since a_{i} ∈ I^{i} and b_{n−i} ∈ I^{n−i} we must have a_{i}b_{n−i} ∈ I^{n} (0 ≤ i ≤ n) and hence e_{n} ∈ I^{n}. Thus c + d ∈ _{B}^{∗}I. It easily follows that _{B}^{∗}I is an additive subgroup of _{B}^{∗}A.

Suppose c ∈ _{B}^{∗}I and d ∈ _{B}^{∗}J. Choose a nonnegative integer p and elements a_{1}, …, a_{p}, b_{1}, …, b_{p} ∈ B such that γ_{n}(a, c) ∈ I^{n} and γ_{n}(b, d) ∈ J^{n} (n ≥ 1), where γ_{n} = γ_{n}(x, z) is as in (2.5). Let F (U, x, z) = 1 + ∑_{i=1}^{p} \binom{U}{i}x_{i}z^{−i} and G(U, y, w) = 1 + ∑_{i=1}^{p} \binom{U}{i}y_{i}w^{−i} where x_{1}, …, x_{p}, y_{1}, …, y_{p}, z, and w are indeterminates. Then F (U, x, z)G(U, y, w) ∈ Q[U, x, y, z^{−1}, w^{−1}]. For integers i and j write i ∧ j for the maximum of {i, j}. Then

F (U, x, z)G(U, y, w) = ∑_{i,j=0}^{p} \binom{U}{i}\binom{U}{j}x_{i}y_{j}z^{i∧j−i}w^{i∧j−j}(zw)^{−(i∧j)},

where x_{0} = y_{0} = 1.

Let t be a new indeterminate and define

H(U, x, y, z, w, t) = ∑_{i,j=0}^{p} \binom{U}{i}\binom{U}{j}x_{i}y_{j}z^{i∧j−i}w^{i∧j−j}t^{−(i∧j)},

where x_{0} = y_{0} = 1. Then H(U, x, y, z, w, t) ∈ Q[U, x, y, z, w, t^{−1}]. Furthermore,

H(0, x, y, z, w, t) = 1;
t^{n}H(n, x, y, z, w, t) ∈ Z[x, y, z, w, t] ∀n ≥ 1; and
(cd)^{n}H(n, a, b, c, d, cd) = γ_{n}(a, c)γ_{n}(b, d) ∈ I^{n}J^{n} = (IJ)^{n} ∀n ≥ 1.

Thus cd ∈ B^{∗}(IJ) by part (9) of (2.10). By letting J = A we see that B^{∗}I is an

ideal of B^{∗}A.

3 The Weak Normalization of the Rees Ring

Again let A ⊆ B be an extension of rings, let I be an ideal of A and let R denote the Rees ring A[It], where t is an indeterminate. Then we have extensions of graded rings A[It] ⊆ A[t] ⊆ B[t]. The weak normalization of A[It] in B[t] is again a graded subring of B[t]; the proof of this fact that is presented below was communicated to us by Leslie Roberts.

Proposition 3.1 Let R ⊆ S be an extension of graded rings. Then the weak normalization of R in S is the filtered union of all subrings of S that can be obtained by a finite sequence of elementary subintegral and elementary weakly subintegral extensions by homogeneous elements. In particular, _{S}^{∗}R is a graded subring of S.

Proof. Let R′ be the union of all extensions of R in S that can be obtained by a finite sequence of extensions of the form T ⊆ T [x], with x ∈ S homogeneous and either x^{2}, x^{3} ∈ T of x^{p}, px ∈ T for some rational prime p. By [6, (2.4)] R′ is seminormal in S and clearly R′ is a graded subring of S.

Just suppose R′ ⊂ _{S}^{∗}R. Then there exists an element s ∈ S − R′ and a rational prime p such that s^{p}, ps ∈ R′. Write s = s_{1} + s_{2} + … + s_{l}, where elements s_{i} are nonzero homogeneous elements with strictly increasing degrees. Choose such an s with l = l(s) as small as possible. Notice that l ≥ 2, since s ∉ R′. The homogeneous components of s^{p} and ps lie in R′ since R′ is graded. Thus s_{1}^{p}, ps_{1}, ∈ R′ and we may conclude s_{1} ∈ R′. Furthermore, if 1 ≤ i ≤ p − 1 and n ≥ p − 1 and we write ni = mp + r with 0 ≤ r ≤ p − 1, then

(ps^{i})^{n} = p^{n}s^{mp+r} = p^{n}s^{r}(s^{p})^{m} = p^{n−r}(ps)^{r}(s^{p})^{m} ∈ R′.

Since R′ is seminormal in S we may deduce that ps^{i} ∈ R′(1 ≤ i ≤ p − 1). Now let t = s − s_{1}. Then

t^{p} = (s − s_{1})^{p} = s^{p} + ∑_{i=1}^{p−1}(−1)^{i}\binom{p}{i}s^{p−i}s_{1}^{i} + s_{1}^{p}.

Now s^{p} ∈ R′ by assumption and s_{1} ∈ R′ by our earlier discussion. The binomial coefficients \binom{p}{i} with 1 ≤ i ≤ p − 1 are divisible by p so the factors \binom{p}{i}s^{p−i} with 1 ≤ i ≤ p − 1 are also in R’ as above. Thus t^{p} ∈ R′. We also have pt = ps − ps_{1} ∈ R′. Since l(t) = l(s − s_{1}) = l(s) − 1, this contradicts our choice of s.

The following lemma connects the notions of weak subintegrality over an ideal and weak subintegrality over a graded ring.

Lemma 3.2 With notation and assumptions as above, an element bt^{m} ∈ B[t] is weakly subintegral over A[It] ⇔ b ∈ B is weakly subintegral over I^{m}.

Proof. Suppose bt^{m} ∈ B[t] is weakly subintegral over A[It]. Choose p ∈ Z^{+} and f_{i} ∈ A[It] (i ≥ 1) with

(bt^{m})^{n} + ∑_{i=1}^{n} \binom{n}{i}f_{i}(bt^{m})^{n−i} = 0 ∀n ≥ p.

For each integer n ≥ p, every homogenous component of the indicated polynomial expression in bt^{m} is zero. Replacing each f_{i} by a_{i}t^{mi}, which is the homogeneous component of f_{i} of degree mi, we get the degree mn homogeneous component of the polynomial expression so that

(bt^{m})^{n} + ∑_{i=1}^{n} \binom{n}{i}(a_{i}t^{mi})(bt^{m})^{n−i} = 0 ∀n ≥ p.

Since a_{i} ∈ (I^{m})^{i} we see that b is weakly subintegral over I.

The converse is clear.

With this we obtain an explicit characterization of the weak normalization of the Rees ring of A over I in terms of the weak subintegral closures of the powers of I.

Theorem 3.3 Let A ⊆ B be an extension of rings, I an ideal of A, let R denote the Rees ring A[It], where t is an indeterminate, and let S = B[t]. Then,

_{S}^{∗}R = ⊕_{n≥0} _{B}^{∗}(I^{n})t^{n}.

In particular, B^{∗}(I^{n}) contains each element of B that is weakly subintegral over

B^{∗}(I^{n}), for each n ≥ 0.

Proof. By (3.1) _{S}^{∗}R is a graded subring of S. Let b ∈ B and n ≥ 0. Then, bt^{n} ∈ _{S}^{∗}R ⇔ bt^{n} is a weakly subintegral over R [13, (6.11)] ⇔ b is weakly subintegral over I^{n} by (3.2) ⇔ bt^{n} ∈ _{B}^{∗}(I^{n})t^{n}. Hence the degree n piece of _{S}^{∗}R is _{B}^{∗}(I^{n})t^{n} for each n and the main assertion of the theorem follows.

The second assertion easily follows from the the fact that the weak normalization of R in S is weakly normal in S, and hence weakly integrally closed in S.

Corollary 3.4 For an ideal I of a ring A we have

∗∗I = ^{∗}I.

Proof. This follows immediately from (3.3) when we take B = A.

In the next result we assume that A is a reduced ring with finitely many minimal primes and that I is a regular ideal of A, that is, I contains a regular element. Let R = A[It] denote the indicated Rees ring. Then R is again reduced with finitely many minimal primes and is an N-graded ring.

Corollary 3.5 Let A be a reduced ring with finitely many minimal primes and I a regular ideal of A. Let Q denote the total quotient ring of A and R = A[It]. Then

∗R = ⊕_{n≥0} _{Q}^{∗}(I^{n})t^{n}.

Proof. Since R is again reduced with finitely many minimal primes it makes sense to talk about the absolute notion of the weak normalization of R. Let K denote the total quotient ring of R. Since I contains a regular element the total quotient ring of A[It] coincides with the total quotient ring of A[t] and hence Q[t] ⊆ K. Since Q[t] is integrally closed in K, we have _{K}^{∗}R =_{Q[t]}^{∗} R. The assertion now follows from the preceding theorem.

We would like to point out that in calculating the weak subintegral closure of an ideal I in an extension B of A we can assume that A is weakly normal in B as the following result illustrates.

Corollary 3.6 Let A ⊆ B be an extension of rings and I an ideal of A. Then,

_{B}^{∗}(I(_{B}^{∗}A)) = _{B}^{∗}I.

Proof. By (2.10) _{B}^{∗}I is ideal of _{B}^{∗}A and hence _{B}^{∗}I contains I(_{B}^{∗}A). Since _{B}^{∗}I is weakly subintegrally closed in B by (3.3) we must have _{B}^{∗}(I(_{B}^{∗}A)) ⊆ _{B}^{∗}I. The opposite containment is clear so the ideals are equal.

4 Examples

For an ideal I of ring A we have a chain of ideals of A:

I ⊆ ^{∗}I ⊆ \overline{I} ⊆ \sqrt{I}.

In this section we give examples that show each of these containments may be proper.

Example 4.1 The integral closure of an ideal I may be properly contained in the radical of I. Let I = (x^{2}) ⊆ k[x], where k is a field and x an indeterminate.

In this case the principal ideal I is integrally closed in k[x] (argue as in (2.2)), and we have I = ^{∗}I = \overline{I} ⊂ \sqrt{I} = (x).

Example 4.2 The weak subintegral closure of an ideal may be properly contained in the integral closure of that ideal. Let k be a field of characteristic zero, u, v indeterminates, and A = k[u, uv, v^{2}] ⊂ k[u, v]. Then A is the affine coordinate ring of the Whitney umbrella and is known to be weakly normal [5,(3.5)]. Let I = (uv). By (2.2) I is weakly subintegrally closed in A. Then a_{2} = v^{2}(uv)^{2} ∈ I^{2} and uv^{2} satisfies the equation X^{2} −a_{2} = 0. Thus uv^{2} ∈ \overline{I} \^{∗}I and hence ^{∗}I ⊂ \overline{I}.

Example 4.3 A nonzero principal ideal of an integral domain A can fail to be weakly subintegrally closed in A when A is not weakly normal. Let A be an integral domain that is not weakly normal, a a nonzero element of A, and let I = (a). Then, I ⊂ ^{∗}I if and only if a ∈ Z(^{∗}A/A), the set of zero divisors for the A-module ^{∗}A/A. To see this notice that for b ∈ A\, b is weakly subintegral over I ⇔ b/a is weakly subintegral over A and b/a ∉ A. In particular, if k is a field, x an indeterminate, A = k[x^{2}, x^{3}], and I = (x^{2}) then x^{3} ∈ ^{∗}I, since x^{3}/x^{2} ∈ ^{∗}A = k[x]. Thus ^{∗}I = (x^{2}, x^{3}), the irrelevant maximal ideal of A.

Example 4.4 The contraction to A of the extension of I to the weak normalization ^{∗}A of A need not equal ^{∗}I. Let A = k[u, uv, v^{2}] ⊂ k[u, v] be as in (4.2) and let I = (u, v^{2}). Notice that (uv)^{2} = u^{2}v^{2} ∈ I^{2}, (uv)^{3} = u^{2}(uv)v^{2} ∈ I^{3}. Thus uv ∈ ^{∗}I so that ^{∗}I = (u, uv, v^{2}). In this case A = ^{∗}A and I(^{∗}A) ∩ A = I ⊂ ^{∗}I.

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