On decomposing ideals into products of comaximal ideals

J. W. Brewer and W. J. Heinzer

Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431 E-mail address: brewer@fau.edu

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 E-mail address: heinzer@math.purdue.edu

I. Introduction

A classical result in commutative ring theory as recorded by W. Krull in [7, page 24] asserts that if D is a one-dimensional Noetherian domain, then each nonzero ideal A of D is a product of pairwise comaximal primary ideals. Another result along these lines is a variant of the Chinese Remainder Theorem: an ideal A of a commutative ring R is a product of pairwise comaximal primary ideals if and only if R/A is a finite direct sum of rings R_{i} where (0) is a primary ideal in each R_{i}. In particular, if dim(R/A) = 0 and A is contained in only

1

2000 Mathmatics Subject Classification Numbers: 13A15, 13C05, 13F05, 13G05.

Keywords and phrases: comaximal ideals, prime radical, primary ideal, Prüfer domain, Dedekind domain.

Copyright ^{_{C} ◯} 2000 by

2 J. W. Brewer and W. J. Heinzer

finitely many prime ideals, the Chinese Remainder Theorem implies that A is a finite product of primary ideals. This leads us to wonder:

Question. Which integral domains D have the property that each nonzero ideal A of D can be written as a product

A = Q_{1} · Q_{2} · · · · · Q_{n}

where the Q_{i}’s are pairwise comaximal and have some additional property?

We also consider this question for nonzero principal ideals aD of D. In the case where aD is a principal ideal, a useful observation is that the factors Q_{i} are invertible ideals.

Obviously, any Dedekind domain has the property; in this case, each Q_{i} is a power of a maximal ideal. In this paper we settle the question in the following cases: (i) each Q_{i} has prime radical (Theorem 1); (ii) each Q_{i} is primary (Theorem 8); and (iii) each Q_{i} is the power of a prime (Theorem 9).

We became interested in this property from reading the paper [11] by Sáez-Schwedt and Sánchez-Giralda. It is well known in the classical theory of linear dynamical systems over fields that canonical forms exist for controllable systems of any dimension. In the paper [2] of J. Brewer and L. Klingler, it is shown by means of representationtheoretic methods, that canonical forms are not likely to exist over arbitrary principal ideal domains. The problem treated in [11] is to try to determine a canonical form for two-dimensional controllable systems over principal ideal domains (and Dedekind domains). In [11], the property above for all nonzero principal ideals was the property the authors needed in order to carry out their successful program.

Questions similar to the ones considered here, but without the co-

On decomposing ideals into products of comaximal ideals 3

maximality, have received attention by others. We thank the referee for suggesting the following papers and the references listed there [1], [8], [9]. We also thank Laszlo Fuchs for several helpful comments and for sending us the article [3] concerning ideal theory in Prüfer domains of finite character. As Professor Fuchs noted when we exchanged articles, our paper contains a result identical with one of the theorems in [3]. We arrived at the theorem from completely different directions.

II. Main Results

We begin with the weakest requirement on the factors and the case of nonzero principal ideals.

Theorem 1 Let D be an integral domain. The following are equivalent:

1. Each nonzero principal ideal aD of D can be written in the form aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

2. D has the following properties:

(a) For any maximal ideal M of D, the set of prime ideals of D contained in M is linearly ordered under inclusion.

(b) Each nonzero principal ideal of D has only finitely many minimal primes.

Proof. (1) ⇒ (2) : Suppose that M is a maximal ideal of D and that P_{1}and P_{2} are incomparable prime ideals of D contained in M. Let a be in P_{1} not P_{2} and b be in P_{2} not P_{1}. The principal ideal abD can be written in the form Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal. Since abD ⊆ M, some Q is contained in M, say Q_{1}. Moreover, since the Q_{i}’s are pairwise

4 J. W. Brewer and W. J. Heinzer

comaximal, only Q_{1} is contained in M. Thus, since P_{1} and P_{2} are contained in M and contain abD, Q_{1} ⊆ P_{1} ∩ P_{2}. But, Q_{1} cannot have prime radical, for suppose that \sqrt{Q_{1}} = P, prime. Then P is contained in P_{1} ∩ P_{2}. Also, P must contain one of a or b. But this is a contradiction to the choice of a and b and it follows that condition (a) holds.

Let a be a nonzero element of D. Write aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical P_{i} and the Q_{i}’s are pairwise comaximal. If P is a minimal prime of aD, then P contains some Q_{i} and hence, P = P_{i}. It follows that {P_{1}, P_{2}, …, P_{n}} is the set of minimal prime ideals of aD. This proves condition (b).

(2) ⇒ (1) : Suppose that aD is a nonzero principal ideal of D and that P_{1}, P_{2}, …, P_{n} are the minimal primes of aD. By condition (a), the P_{i}’s, are pairwise comaximal. Hence, there exist x_{i} in P_{i} and y_{i} in P_{j} for j ≠ i such that x_{i} + y_{i} = 1. Let D_{i} = D[1/y_{i}], the localization of D at the powers of the element y_{i}. If Q_{i} = aD_{i} ∩ D, then Q_{i} has radical P_{i}. Let M be a maximal ideal of D. Then either aD ⊆ M or aD ⊈ M. If aD ⊈ M, then aD_{M} ∩ D = D. If aD ⊆ M, then M ⊃ P_{i} for some i, and hence y_{i} ∉ M; it follows that D_{M} is a term in the intersection D[1/y_{i}] = ⋂ D_{P}, where P is prime and y_{i} ∉ P. Thus, D ⊆ ⋂_{i=1}^{n} D_{i} ∩ D[1/a] ⊆ ⋂_{M max} D_{M} = D. Now, for every ideal A of

D, A = ⋂_{M max} AD_{M} ∩ D. It follows that aD = Q_{1} ∩ Q_{2} ∩ ⋯ ∩ Q_{n}.

Moreover the Q_{i}’s are pairwise comaximal, so their intersection is their product. This completes the proof.

The proof of Theorem 1 can be easily modified to prove

Theorem 2 Let D be an integral domain. The following are equivalent:

1. Each nonzero ideal A of D can be written in the form A =

On decomposing ideals into products of comaximal ideals 5

Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

2. D has the following properties:

(a) For any maximal ideal M of D, the set of prime ideals of D contained in M is linearly ordered under inclusion.

(b) Each nonzero ideal of D has only finitely many minimal primes.

Remark 3 Note that condition 2(a) above is equivalent to the following: If P and Q are nonzero prime ideals of D, then either P and Q are comaximal or P and Q are comparable. Viewed in this way, conditions 2(a) and 2(b) are topological conditions on the prime spectrum of D with the Zariski topology. More specifically, 2(a) says that any two proper closed irreducible subsets of Spec(D) are either comparable or disjoint; 2(b) says that each closed subset of Spec(D) has only finitely many irreducible components. The paper [5] of M. Hochster establishes the existence of many examples of integral domains having these two properties.

There are situations in which Theorem 1 and Theorem 2 can be combined into one result as the following proposition illustrates.

Proposition 4 Suppose that the integral domain D has the property that each nonzero prime ideal of D is contained in only finitely many maximal ideals. The following are equivalent:

1. Each nonzero ideal A of D can be written in the form A = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

2. Each nonzero principal ideal aD of D can be written in the form aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

6 J. W. Brewer and W. J. Heinzer

3. D has the following properties:

(a) For any maximal ideal M of D, the set of prime ideals of D contained in M is linearly ordered under inclusion.

(b) Each nonzero principal ideal of D has only finitely many minimal primes.

Proof. That (1) implies (2) is obvious and that (2) is equivalent to (3) is the content of Theorem 1. Thus, we have only to prove that (3) implies (1). Since condition (2a) is the same in both theorems, we have to verify that each nonzero ideal A of D has only finitely many minimal primes. Pick a nonzero element a ∈ A and let P_{1}, P_{2}, …, P_{n} be the minimal primes of aD. By hypothesis, each P_{i} is contained in only finitely many maximal ideals, say P_{i} is contained in M_{i1}, …, M_{ik_{i}}. Let P be a minimal prime of A. Then aD ⊆ P and P ⊇ P_{j} for some j. It follows that P ⊆ M_{jl} for some l between j1 and jk_{j}. (P might be contained in more that one such M_{jl}, but that causes no difficulty.) If Q is another minimal prime of A that contains P_{j}, then Q ⊆ M_{ji} for i ≠ l since the primes contained in a given maximal ideal are comparable. Therefore, there are at most k_{j} distinct minimal primes of A that contain P_{j}. Since each minimal prime of A must contain some P_{j}, it follows that there are at most k_{1} + k_{2} + ⋯ k_{n} minimal primes of A.

We record the following corollary to Theorem 2.

Corollary 5 Let D be a Prüfer domain. If each nonzero element of D belongs to only finitely many maximal ideals of D, then each nonzero ideal of D can be written in the form Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

Proof. If D is a Prüfer domain, then each localization of D at a prime ideal is a valuation domain. In particular, the prime ideals of

On decomposing ideals into products of comaximal ideals 7

D contained in a given maximal ideal of D are linearly ordered under inclusion. Let A be a nonzero ideal of D with P a minimal prime of A. Since each nonzero element of D belongs to only finitely many maximal ideals, the same is true for A. Let those maximal ideals be M_{1}, M_{2}, …, M_{k}. Then P ⊆ M_{j} for some j. Since the prime ideals of D contained in a given maximal ideal of D are linearly ordered, any M_{i} can contain at most one minimal prime of A. Thus, A has at most k minimal primes and the result follows from Theorem 2.

Remark 6 Laszlo Fuchs has pointed out to us that the proof of Corollary 5 also applies for a non-Prüfer domain D: if each nonzero element of D belongs to only finitely many maximal ideals and if the prime ideals contained in a given maximal ideal of D are linearly ordered under inclusion, then each nonzero ideal of D can be written in the form Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal.

There exist Prüfer domains which satisfy the hypothesis of Theorem 1 and which have nonzero elements contained in infinitely many maximal ideals. For example, if ℚ denotes the field of rational numbers, ℤ the ring of integers,and X an indeterminate, let M be the maximal ideal Xℚ[X]_{(X)} of ℚ[X]_{(X)}. Then the domain D = ℤ + M is a two-dimensional Prüfer domain having a unique prime ideal M of height one and having infinitely many maximal ideals corresponding to the maximal ideals of ℤ. The elements of M are contained in infinitely many maximal ideals of D while the elements of D not in M are in only finitely many prime ideals. Thus, D satisfies the hypothesis of Theorem 1

We next present an example to illustrate the fact that Theorems 1 and 2 cannot always be combined. Thus, for the case when the ideals Q_{i} are only assumed to have prime radical, the condition for all

8 J. W. Brewer and W. J. Heinzer

principal ideals is not equivalent to the condition for all ideals. More specifically, we construct a Prüfer domain D having the following two properties: Each nonzero principal ideal of D has only finitely many minimal primes; D has a nonzero ideal having infinitely many minimal primes. Since a Prüfer domain has the property that any two prime ideals contained in a given maximal ideal are comparable, this example effectively separates Theorem 1 from Theorem 2. Thus, for this Prüfer domain D, each nonzero principal ideal aD can be written in the form aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} has prime radical and the Q_{i}’s are pairwise comaximal, but there exists a nonzero ideal A of D which cannot be so written.

Example 7 We are going to construct a Prüfer domain D having for each positive integer n precisely two primes of height n, M_{n} which is maximal, and P_{n} which is nonmaximal. This gives rise to the diagram below describing the prime ideal lattice of D (the union of the P_{n} is a unique maximal ideal of infinite height).

⋮  ⋮
M_{4}  P_{4}
↖ ↑
M_{3}  P_{3}
↖ ↑
M_{2}  P_{2}
↖ ↑
M_{1}  P_{1}

↖↗ (0)

We construct D to be the union of a chain of Prüfer subdomains D_{n}, where D_{n} is n-dimensional and D_{n} has two maximal ideals of height n and has for each positive integer m < n two primes of height

On decomposing ideals into products of comaximal ideals 9

m, exactly one of which is maximal. The construction is such that there exists an inclusion map of D_{n} into D_{n+1} so that the two maximal ideals of D_{n+1} of height n + 1 both lie over the same maximal ideal of D_{n}. Moreover there is also a nonmaximal prime of D_{n+1} of height n lying over this same maximal ideal of D_{n}. For each m < n, the maximal ideal of D_{n+1} of height m lies over the maximal ideal of D_{n} of height m and the nonmaximal prime of D_{n+1} of height m lies over the nonmaximal prime of D_{n} of height m.

Let A := ∩_{n=1}^{∞}M_{n}. We show that each of the M_{n}’s is a minimal prime of A, so that A has infinitely many minimal primes. This follows from the fact that A_{n} := A ∩ D_{n} is the Jacobson radical of D_{n} and D_{n} has precisely n + 1 maximal ideals, one of height i for each i between 1 and n − 1 and two of height n. We have A_{n}D_{n+1} properly contained in A_{n+1} and A = ∪_{n=1}^{∞}A_{n}. As we go up from D_{n} to D_{n+1}, our ideal A is picking up more minimal primes and in the union has infinitely many. Now each principal ideal of D has the form aD, where a is in D_{n} for some n. Then aD_{n} has at most n + 1 minimal primes and by the construction, aD has at most n + 1 minimal primes. The point is that in passing from D_{n} to D_{n+1}, there are three prime ideals of D_{n+1} lying over the same maximal ideal of height n of D_{n}. If a is contained in one of them, then it is in all three and has precisely one of them as a minimal prime.

We now spell out a way to construct such a Prüfer domain D. (We will be intersecting finite families of valuation domains, and using the theorem on independence of valuations as given by Nagata in [10, (11.11), page 38].) For k a field and x_{1}, x_{2}, … indeterminates over k, we construct a chain D_{n} of the type discussed above, where D_{n} has fraction field k(x_{1}, …, x_{n}) : Let D_{1} be the intersection of the

10 J. W. Brewer and W. J. Heinzer

DVRs (k[x_{1}])_{(x_{1})} = V_{1} and (k[x_{1}])_{(x_{1}−1)} = W_{1}. Let

M_{11} = (x_{1} − 1)D_{1} and P_{11} = x_{1}D_{1}.

denote the maximal ideals of D_{1}. We construct D_{2} on the field k(x_{1}, x_{2}) as an intersection of three valuation domains. We denote by W_{1}(x_{2}) the valuation domain (k(x_{2})[x_{1}])_{(x_{1}−1)}, i.e., W_{1}[x_{2}] localized at the extension of the maximal ideal of W_{1} to this polynomial ring. This valuation domain W_{1}(x_{2}) is sometimes called the Gaussian or trivial extension of W_{1} to the simple transcendental field extension generated by x_{2}. We define valuation domains W_{2} and V_{2} of rank 2 on k(x_{2}, y_{2}) both of which extend V_{1}. Let N_{2} denote the maximal ideal of V_{1}(x_{2}) and let

V_{2} = (k[x_{2}])_{(x_{2})} + N_{2} and W_{2} = (k[x_{2}])_{(x_{2}−1)} + N_{2}.

Define D_{2} := W_{1}(x_{2}) ∩ W_{2} ∩ V_{2}. Using [10, (11.11)], we see that D_{2} has two maximal ideals of height 2,

M_{22} := (x_{2} − 1)D_{2} and P_{22} := x_{2}D_{2},

and two prime ideals of height one, M_{21} := (x_{1} − 1)D_{2} which is maximal, and P_{21} := N_{2} ∩ D_{2} which contains x_{1} and is not maximal. Observe that M_{22}, P_{22} and P_{21} all have the property that their intersection with D_{1} is P_{11}.

We construct D_{3} on k(x_{1}, x_{2}, x_{3}) as an intersection of four valuation domains. Let N_{3} denote the maximal ideal of V_{2}(x_{3}). Define

V_{3} := (k[x_{3}])_{(x_{3})} + N_{3} and W_{3} = (k[x_{3}])_{(x_{3}−1)} + N_{3}.

Define D_{3} := W_{1}(x_{2}, x_{3}) ∩ W_{2}(x_{3}) ∩ W_{3} ∩ V_{3}. Using [10, (11.11)], we see that D_{3} has two maximal ideals of height 3,

M_{33} := (x_{3} − 1)D_{3} and P_{33} := x_{3}D_{3},

two prime ideals of height two, M_{32} := (x_{2} − 1)D_{3} which is maximal,

On decomposing ideals into products of comaximal ideals 11

and P_{32} := N_{3} ∩ D_{3} which contains x_{2} and is not maximal, and two prime ideals of height one, M_{31} := (x_{1} − 1)D_{3} which is maximal and P_{31} which is the contraction to D_{3} of the height-one prime of V_{3}.

Let t ≥ 3 be a fixed positive integer and assume for each positive integer n ≤ t we have constructed n + 1 valuation domains

W_{1}(x_{2}, …, x_{n}), W_{2}(x_{3}, …, x_{n}), …, W_{i}(x_{i+1}, …, x_{n}), …, W_{n} and V_{n}

on the field k(x_{1}, …, x_{n}) such that W_{m} has rank m for each m with 1 ≤ m ≤ n, V_{n} has rank n, and

D_{n} := W_{1}(x_{2}, …, x_{n}) ∩ W_{2}(x_{3}, …, x_{n}) ∩ ⋯ ∩ W_{n} ∩ V_{n}

has the following properties:

1. D_{n} has two maximal ideals of height n, M_{nn} = (x_{n} − 1)D_{n} and P_{nn} = (x_{n})D_{n}.

2. For each positive integer m < n, D_{n} has two prime ideal of height m, M_{nm} = (x_{m} − 1)D_{n}, which is maximal, and P_{nm} which is nonmaximal.

3. For positive integers m ≤ n ≤ s ≤ t, the prime ideal P_{sm} of D_{s} intersects D_{n} in P_{nm}.

It is clear that we can continue the construction by defining t + 2 valuation domains on k(x_{1}, …, x_{t+1}) as follows. Let N_{t+1} denote the maximal ideal of V_{t}(x_{t+1}) and define V_{t+1} = k[x_{t+1}]_{(x_{t+1})} + N_{t+1} and W_{t+1} = k[x_{t+1}]_{(x_{t+1}−1)} + N_{t+1}. For 1 ≤ i ≤ t, define the Gaussian extension W_{i}(x_{i+1}, …, x_{t+1}). Then D_{t+1} defined as the intersection of these t + 2 valuation domains on k(x_{1}, …, x_{t+1}) gives the properties listed above for positive integers m ≤ n ≤ s ≤ t + 1. Therefore, by induction, we have the stated properties for all positive integers m ≤ n ≤ s.

We define D = ∪_{n=1}^{∞}D_{n}. For each positive integer m, D has two prime ideals of height m, M_{m} := ∪_{n=m}^{∞}M_{nm}, which is maximal, and

12 J. W. Brewer and W. J. Heinzer

P_{m} := ∪_{n=m}^{∞}P_{nm}, which is nonmaximal. D also has a unique maximal ideal M_{∞} := ∪_{n=1}^{∞}P_{nn} having infinite height. This completes the construction of the example.

By [10, (11.11)], each of the D_{n} is a Bezout domain. Therefore D is also a Bezout domain.

If we strengthen the assumption on the factors, we get stronger conditions on the domain. In addition, we also get that the condition on all nonzero ideals is equivalent to the condition on all principal ideals.

Theorem 8 Let D be an integral domain. The following are equivalent:

1. Each nonzero principal ideal aD of D can be written in the form aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} is primary and the Q_{i}’s are pairwise comaximal.

2. D has the following properties:

(a) D is one-dimensional.

(b) D has Noetherian spectrum.

3. Each nonzero ideal A of D can be written in the form A = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} is primary and the Q_{i}’s are pairwise comaximal.

Proof. (1) ⇒ (2): Let P be a nonzero prime ideal of D. If P is not maximal, let M be a maximal ideal of D that properly contains P and let a be a nonzero element of P. Let b be an element of M not in P. Set x = ab. By hypothesis, xD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} is primary and the Q_{i}’s are pairwise comaximal. Now, P contains x and so P contains some Q, say Q_{1}. Let P_{1} = \sqrt{Q_{1}}. Thus Q_{1} is P_{1}- primary and P_{1} ⊆ P. Also, because the Q_{i}’s are pairwise comaximal, M contains at most one minimal prime ideal of xD and that minimal

On decomposing ideals into products of comaximal ideals 13

prime must be P_{1}. Localize at the maximal ideal M. This forces xD_{M} = Q_{1}D_{M} to be P_{1}D_{M}-primary. But this is impossible, for b is not in P_{1}D_{M}, but ab = x is in xD_{M}. If xD_{M} were P_{1}D_{M}-primary, then a must be in xD_{M}. But a = x(\frac{d}{s}), for some d ∈ D, s ∈ D \ M implies x = x(\frac{d}{s})b, so s = db. This contradicts the fact that b is in M. Hence, D is one dimensional.

To see that D has Noetherian spectrum, we first show that each nonzero ideal of D is contained in only finitely maximal ideals. Let a be a nonzero element of D. Then we can write aD = Q_{1} · Q_{2} · · · · · Q_{n}, where \sqrt{Q_{i}} = M_{i} is prime; in fact, M_{i} is a maximal ideal since D is one-dimensional. If M is any maximal ideal of D of which a is a member, then M ⊇ M_{j} for some j. Thus, {M_{1}, M_{2}, …, M_{n}} is the set of maximal ideals of D containing a. If A is any nonzero ideal of D, then the set of maximal ideals containing A is a subset of the set of maximal ideals containing any nonzero element inside A. In particular, the set of maximal ideals containing A is finite.

To show that D has Noetherian spectrum,we must prove that D satisfies the ascending chain condition on radical ideals. Thus, let A_{1} ⫋ A_{2} ⫋ … ⫋ be an ascending sequence of radical ideals. Now, \sqrt{A_{1}} = M_{1} ∩ M_{2} ∩ ⋯ ∩ M_{k} for some maximal ideals M_{1}, M_{2}, …, M_{k} of D. Since A_{1} ⫋ A_{2}, the set of maximal ideals of D that contain A_{2} is a proper subset of {M_{1}, M_{2}, …, M_{k}}. It follows that the sequence must be finite.

(2) ⇒ (3) : Let A be a nonzero ideal of D, with M_{1}, M_{2}, … M_{n} the maximal ideals containing A. That there are only finitely many follows from the fact that D is one-dimensional and the fact that in a ring with Noetherian spectrum, each nonzero ideal has only finitely many minimal primes. Set Q_{i} = AD_{M_{i}} ∩ D. Then Q_{i} has radical M_{i} and A is the intersection of the Q_{i}’s. Since the M_{i}’s are maximal, the Q_{i}’s are primary and comaximal. So, A is also the product of the Q_{i}’s.

(3) ⇒ (1) : This is obvious.

Finally, we treat the case where the Q’s are prime powers.

Theorem 9 Let D be an integral domain. The following are equivalent:

1. Each nonzero principal ideal aD of D can be written in the form aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} is a prime power and the Q_{i}’s are pairwise comaximal.

2. D has the following properties:

(a) D is one-dimensional.

(b) D is Noetherian.

(c) D is integrally closed.

3. Each nonzero ideal A of D can be written as a product of prime ideals.

4. D is a Dedekind domain.

5. Each nonzero ideal A of D can be written in the form A = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} is a prime power and the Q_{i}’s are pairwise comaximal.

Proof. The equivalence of properties (2) - (5) is standard, see for example [4, Theorem 37.8]. In particular, (3) ⇒ (4) is a theorem of Matsusita [6, Page 68]. It is obvious that (5) ⇒ (1) Thus it suffices to show:

(1) ⇒ (2) : We first prove that D is one-dimensional. Let M be a maximal ideal of D and suppose there exists a nonzero prime ideal P properly contained in M. Let x ∈ M, x ∉ P and write xD = P_{1}^{e_{1}} ⋯ P_{n}^{e_{n}}, where the P_{i} are pairwise comaximal prime ideals. It follows that the P_{i} are invertible and M contains one, and only one, of the P_{i}, say P_{1}. Because x ∉ P, P_{1} is not contained in P. Let y be a nonzero element of P and write yD = N_{1}^{f_{1}} ⋯ N_{m}^{f_{m}}, where the N_{j} are pairwise comaximal prime ideals. Thus the N_{j} are invertible and P contains one, and only one, of the N_{j}, say N_{1}. By Theorem 1, the prime ideals contained in M are linearly ordered with respect to inclusion. Since N_{1} ⊆ P, we have N_{1} ⊊ P_{1}. It follows that N_{1}D_{P_{1}} ⊊ P_{1}D_{P_{1}} are nonzero principal prime ideals of D_{P_{1}}. This is a contradiction. (Suppose R is a quasilocal domain with principal maximal ideal xR and that yR is a principal prime ideal with yR strictly smaller than xR. Write y = xz for some z ∈ R. Since yR is a prime ideal and x ∉ yR, it follows that z ∈ yR, say z = yr. Then y = xyr, so y(1 − xr) = 0. Since 1 − xr is a unit of R, y = 0).

Let M be a maximal ideal of D and let a be a nonzero element of M. Then aD = Q_{1} · Q_{2} · · · · · Q_{n}, where each Q_{i} = M_{i}^{e_{i}} for some maximal ideal M_{i} of D. Thus, M = M_{j} for some j. Since a product of ideals is invertible if and only if each factor is invertible, it follows that M is invertible. In particular, M is finitely generated. So, all prime ideals of D are finitely generated and D is Noetherian by Cohen’s Theorem [6, Theorem 8]. Finally, MD_{M} is principal since M is invertible. Therefore, each prime ideal of the quasi-local ring D_{M} is principal and by an analog to Cohen’s Theorem, [6, Page 8] D_{M} is a principal ideal domain. It follows that D is integrally closed.

This paper was written, composed, and typeset electronically. The difficulty posed by two authors working in different locations and using different interfaces to Tex would have been insurmontable without the help of Fred Richman and to Fred we offer our sincere thanks.

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