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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2017; 57(3): 433-439

Published online September 23, 2017

### A Characterization of Dedekind Domains and ZPI-rings

Esmaeil Rostami

Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

Received: February 3, 2017; Accepted: September 9, 2017

### Abstract

It is well known that an integral domain D is a Dedekind domain if and only if D is a Noetherian almost Dedekind domain. In this paper, we show that an integral domain D is a Dedekind domain if and only if D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology. We also give a new characterization of ZPI-rings.

Keywords: Dedekind domain, almost Dedekind domain, ZPI-ring

### 1. Introduction

Let R denote throughout a commutative ring with 1. Recall that a ring R is called a ZPI-ring if every proper ideal of R is a product of prime ideals. Also, an integral domain D is called a Dedekind domain if every proper ideal of D is a product of prime ideals. It is well known that if D is a Dedekind domain, then is a Noetherian valuation domain for each maximal ideal of D and the converse is true if D is Noetherian. An integral domain D is called almost Dedekind if is a Noetherian valuation domain for each maximal ideal of D. Thus, a Dedekind domain is an almost Dedekind domain and a Noetherian almost Dedekind domain is a Dedekind domain. In [6], Loper discussed methods for constructing nonNoetherian almost Dedekind domains.

Almost Dedekind domains and ZPI-rings have many of the important properties of Dedekind domains that make them two useful and attractive classes of rings. We give below some results which demonstrates some of the properties of Dedekind domains, almost Dedekind domains and ZPI-rings.

### Theorem 1.1. ([5, Theorem 6.20])

If D is a Noetherian integral domain, then the following statements are equivalent:

• D is a Dedekind domain.

• D is integrally closed and every nonzero proper prime ideal of D is maximal.

• Every nonzero ideal of D generated by two elements is invertible.

• If AB = AC, where A, B, C are ideals of D and A ≠ 0, then B = C.

• For every maximal idealof D, the ring of quotientsis a valuation ring.

### Theorem 1.2. ([5, Theorem 9.4])

If D is an integral domain which is not a field, then the following statements are equivalent:

• D is an almost Dedekind domain.

• D has Krull dimension one and each primary ideal of D is a power of its radical.

### Theorem 1.3. ([5, Theorem 9.10])

The following statements are equivalent for a ring R.

• R is a ZPI-ring.

• R is a Noetherian ring such that for each maximal idealof R, there are no ideals of R strictly betweenand.

• R is a direct sum of a finite number of Dedekind domains and special primary rings.

For more information about Dedekind domains, almost Dedekind domains and ZPI-rings, refer to [4] and [5].

In this paper by using Max(D) as a subspace of Spec(D) with respect to the Zariski topology, we show that an integral domain D is a Dedekind domain if and only if D is an almost Dedekind domain such that Max(D) is a Noetherian topological space. We also give a new characterization of ZPI-rings.

### 2. Main Results

Recall that a ring R is said to be indecomposable, if R cannot be written as R = R1 × R2, where R1 and R2 are both nonzero rings. It is well known, and not difficult to prove, that a ring R is indecomposable if and only if 1 is the only nonzero idempotent of R.

### Lemma 2.1

The following statements are equivalent for an ideal I in R.

• The ideal I can be written as I = J1J2 ∩ … ∩ Jn, where J1, J2,,Jnare ideals in R such that each of the R/Jiis indecomposable.

• The ideal I can be written as I = K1K2Km = K1K2∩…∩Km, where K1, K2,,Kmare pairwise comaximal ideals in R such that each of the R/Kiis indecomposable.

• R/I has only finitely many idempotents.

Proof

(1) ⇒ (2) Let I can be written as I = J1J2∩…∩Jn, where J1, J2,…,Jn are ideals in R such that each of the R/Ji is indecomposable. Let two of the Ji’s, say J1 and J2, are contained in a maximal ideal of R. If e+(J1J2) is an idempotent in R/(J1J2), then e+J1 and e+J2 are idempotents in R/J1 and R/J2, respectively. Now since R/J1 and R/J2 are indecomposable and J1 and J2, we have e + (J1J2) = 0+(J1J2) or e + (J1J2) = 1+(J1J2). Hence, R/(J1J2) is also indecomposable. Set I1=(J1J2). Hence, I=I1I3Jn such that R/I1, R/J3,, R/Jn are indecomposable. Repeating this argument, we see that the ideal I can be written as I = K1K2 ∩ … ∩Km, where K1, K2,…,Km are pairwise comaximal ideals of R such that each of the R/Ki is indecomposable. Now since K1,K2,…,Km are pairwise comaximal, we have K1K2Km = K1K2 ∩ … ∩ Km. (2) ⇒ (3) Let the ideal I can be written as I = K1K2Km = K1K2 ∩ … ∩ Km, where K1, K2,…,Km are pairwise comaximal ideals of R such that each of the R/Ki is indecomposable. By the Chinese Remainder Theorem, R/Ii=1mR/Ki. Now since each of the R/Ki has no nontrivial idempotents, R/I has only finitely many idempotents.

(3) ⇒ (1) If R/I is indecomposable, there is nothing to prove. Suppose that R/I is not indecomposable. Then there exists a nontrivial idempotent r + I in R/I. Thus, {0 + I} = I/I = (〈I, r〉/I) ∩ (〈I, r − 1〉/I) = (〈I, r〉/I)(〈I, r − 1 〉/I). Hence, I = 〈I, r〉 ∩ 〈I, r − 1〉 and R/IR/〈I, r〉 ⊕ R/〈I, r − 1〉. Now if R/〈I, r〉 and R/〈I, r − 1〉 are indecomposable, the proof is complete. Otherwise, R/〈I, r〉 or R/〈I, r−1〉 can be written as a direct sum of nonzero rings as above. Since R/I has finitely many idempotents, this process terminates after finite steps. This completes the proof.

For a ring R, let Spec(R) and Max(R) denote the set of all prime ideals and all maximal ideals of R, respectively. The Zariski topology on Spec(R) is the topology obtained by taking the collection of sets of the form (I) = {P ∈ Spec(R) | IP} (resp. (I) = {P ∈ Spec(R) | IP}), for every ideal I of R, as the open (resp. closed) sets. When considering as a subspace of Spec(R), Max(R) is called MaxSpectrum of R. So, its closed and open subsets are D(I) = (I)∩Max(R) = { ∈ Max(R) | I} and V(I) = (I) ∩ Max(R) = { ∈ Max(R) | I}, respectively.

A topological space X is called Noetherian if every nonempty set of closed subsets of X, ordered by inclusion, has a minimal element. An ideal I of R is called a J-radical ideal if it is an intersection of maximal ideals. Clearly, J-radical ideals of R correspond to closed subsets of Max(R), and Max-Spectrum of R is Noetherian if and only if R satisfies the ascending chain condition for J-radical ideals (See [7] for more details).

### Theorem 2.2

Let R be a ring such that Max(R) is a Noetherian topological space as a subspace of Spec(R) with respect to the Zariski topology. Then every proper ideal I of R can be written as I = J1J2 ∩ … ∩ Jn, where J1, J2,,Jnare ideals of R such that each of the R/Jiis indecomposable.

Proof

Let I be a proper ideal of R. Since Max(R) is Noetherian, Max(R/I) is also Noetherian. Thus, it is sufficient to show that the result is true for the zero ideal. Suppose, on the contrary, that the zero ideal cannot be written as I = J1J2∩…∩Jn, where J1, J2,…,Jn are ideals of R such that each of the R/Ji is indecomposable. By Lemma 2.1, R has infinitely many distinct idempotents, say α1, α2,…,αn,…, and so f1 = α1, f2 = α1α2,…, fn = α1α2αn,…, are infinitely many distinct idempotents in R with fifi+1 = fi+1. Set ei = fifi+1 for each i ∈ ℕ. It is easily seen that {ei}i=1 is an infinite set of nonzero orthogonal idempotents. For each i ∈ ℕ, as 1 − ei ≠ 1, there exists ∈ Max(R) such that 1 − ei.

Set Jn = ∩i∈ℕ{1,2,…,n} for each n ∈ ℕ. Thus J1J2J3 ⊆ …, is an ascending chain of J-radical ideals of R. By hypothesis, there exists k ∈ ℕ such that Jk = Jk+1. Therefore, Jk+1 = ∩i∈ℕ{1,2,…,k+1}Jk = ∩i∈ℕ{1,2,…,k}. Now for all ik + 1, ek+1 = ek+1 − 0 = (1 − ei)ek+1. Thus, ek+1Jk+1. Therefore, ek+1 and 1 − ek+1, a contradiction.

We will need the following well known fact about indecomposable rings which is a consequence of [1, Proposition 27.1].

### Lemma 2.3

Let I be a proper ideal of R. Then R/I is indecomposable if and only if R/I is indecomposable.

An ideal I of a ring R is called semi-primary if I is a prime ideal. In [3] and [2], Gilmer considered rings whose semi-primary ideals are primary. Before proceeding, we state some useful results.

### Theorem 2.4. ([2, Theorem 2])

Let R be a ring whose semi-primary ideals are primary. If Q is a P-primary ideal of R where P is a nonmaximal prime ideal of R, then Q = P.

By Lemma 2.3, if I is a semi-primary ideal, then R/I is indecomposable. The next result is a consequence of Theorem 2.4.

### Corollary 2.5

Let R be a ring with the property that if R/I is indecomposable for an ideal I of R, then I is primary. Then if Q is a P-primary ideal of R where P is a nonmaximal prime ideal of R, then Q = P.

### Lemma 2.6

Let R be a one-dimensional ring such that for any chain Pof prime ideals of R and pP, there exists msuch that p = pm. Then if P is a nonmaximal prime ideal of R and I is an ideal of R with I=P, then I = P.

Proof

Let P be a nonmaximal prime ideal of R, and let I be an ideal of R with I=P. Consider the ring = R/I, and write “bar” for the quotient map. Since I=P, is the unique nonmaximal prime ideal of . Let . If is a maximal ideal of , then . By hypothesis, there exists such that p¯=pm¯. Now let χ̄ be an element of such that χ̄. Then (x¯-xm¯)p¯=0¯. Since x¯-xm¯m¯, the annihilator of must be . Hence, p¯=pm¯=0¯. Therefore, I = P.

### Corollary 2.7. ([3, Corollay 2.2])

Let R be a ring whose semi-primary ideals are primary. Then R has dimension less than two.

### Remark 2.8

It is easily seen that if an ideal I of a ring R can be generated by a set of idempotents, then every element of I is a multiple of an idempotent of I.

### Lemma 2.9

Let I be a ideal of R such that R/I is indecomposable, and let I = 〈{eI | e2 = e}〉. Then R/Iis also indecomposable.

Proof

Let x2+I′ = x+I′ for some xR. Thus, x2xI′ ⊆ I and so x2+I = x+I. Since R/I is indecomposable and x + I is an idempotent element in R/I, we have xI or x−1 ∈ I. Suppose that xI. Nowsince x2xI′, by Remark 2.8, there exists e2 = eI′ such that x2x = re for some rR. Thus, x2x = (x2x)e. Hence, (1−e)x2 = (1−e)x. Thus, ((1−e)x)2 = (1−e)2x2 = (1−e)x2 = (1−e)x. This shows that (1 − e)x is an idempotent element in I, hence (1 − e)xI′. Now since eI′, we have xI′. A similar argument works when x − 1 ∈ I. Therefore, R/I′ has no nontrivial idempotents, and so R/I′ is indecomposable.

### Lemma 2.10

Let R be a zero-dimensional ring. If {Pα}α∈Λis a family of prime ideals of R such that R/∩α∈ΛPαis indecomposable, then | Λ |= 1.

Proof

Let {Pα}α∈Λ be a family of prime ideals of R such that R/∩α∈ΛPα is indecomposable. Since ∩α∈ΛPα is a radical ideal, R/∩α∈ΛPα is a zero-dimensional reduced ring, and so R/∩α∈ΛPα is a V on Neumann regular ring. Suppose r + ∩α∈ΛPα is a nonzero element of R/∩α∈ΛPα, if r + ∩α∈ΛPα is nonunit, then there exists s+∩α∈ΛPα in R/∩α∈ΛPα such that r+∩α∈ΛPα = rsr+∩α∈ΛPα. It is easily seen that sr+∩α∈ΛPα is a nontrivial idempotent of R/∩α∈ΛPα. So R/∩α∈ΛPα is not indecomposable, a contradiction. Thus, every nonzero element of R/∩α∈ΛPα is unit, and so R/∩α∈ΛPα is a field. Thus, | Λ |= 1.

### Proposition 2.11

The following statements are equivalent for a ring R.

• For an ideal I of R if R/I is indecomposable, then I is primary.

• R is a zero-dimensional ring or R is a one-dimensional ring such that every nonmaximal prime ideal of R can be generated by its idempotents.

Proof

(1) ⇒ (2) By Corollary 2.7 and the fact that R/I is indecomposable for every semi-primary ideal I, R has dimension less than two. Let R be a one-dimensional ring, and let P be a nonmaximal prime ideal of R. By Lemma 2.9, R/P is indecomposable, where P = 〈{eP | e2 = e}〉. By hypothesis, P is primary. Since P is a minimal prime ideal over P′, P′ is P-primary ideal. Thus P′ = P by Corollary 2.5. Therefore, every nonmaximal prime ideal of R can be generated by its idempotents. (2) ⇒ (1) Let R be a zero-dimensional ring, and let R/I be indecomposable for an ideal I of R. By Lemma 2.3, R/I=R/IPSpec(R)P is indecomposable. Thus, by Lemma 2.10, I=IPSpec(R)P must be a maximal ideal of R. Hence, I is primary in this case.

Now let R be one-dimensional, and let R/I be indecomposable for an ideal I of R. By Lemma 2.3, R/I=R/IPSpec(R)P is indecomposable. If R/∩ IP∈Spec(R) is a zero-dimension ring, as above, I=IPSpec(R)P is a maximal ideal of R, and so I is primary.

Now let R/∩ IP∈Spec(R)P be a one-dimension ring. We now consider the cases ∩IP∈Spec(R)P is a prime ideal and ∩IP∈Spec(R)P is not a prime ideal.

Case 1

If ∩IP∈Spec(R)P is a prime ideal of R, then ∩IP∈Spec(R)P is a nonmaximal prime ideal of R. If p ∈ ∩IP∈Spec(R)P, by hypothesis and Remark 2.8, there exists an idempotent e ∈ ∩IP∈Spec(R)P such that p = re for some rR. Hence, p = pen for each n ∈ ℕ. Since I=IPSpec(R)P, thus a power of e is in I. Hence, pI, and so ∩IP∈Spec(R)P = I. Therefore, I is primary.

Case 2

If ∩IP∈Spec(R)P is not a prime ideal, there exists a non-maximal prime ideal P1 of a ring R containing ∩IP∈Spec(R)P. By hypothesis, there exists a non-trivial idempotent eP1IP∈Spec(R)P. Thus the ring R/∩IP∈Spec(R)P has a nontrivial idempotent, and so R/∩IP∈Spec(R)P is not indecomposable, a contradiction.

Now we can state the main results of this paper.

### Theorem 2.12

The following statements are equivalent for a ring R.

• R is a ZPI-ring.

• R satisfies the following conditions:

• The dimension of R is at most one and every nonmaximal prime ideal of R can be generated by its idempotents.

• Each primary ideal of R is a power of its radical.

• Max(R) is a Noetherian topological space as a subspace of Spec(R) with respect to the Zariski topology.

Proof

By Theorem 1.3 and the definition of ZPI-rings, it is sufficient to prove (2) ⇒ (1). By Theorem 2.2 and Lemma 2.1, every proper ideal I of R can be written as I = K1K2Km = K1K2 ∩ … ∩Km, where K1, K2,…,Km are pairwise comaximal ideals in R such that each of the R/Ki is indecomposable. Proposition 2.11 implies that each of the Ki is primary, and so every proper ideal of R can be written as a product of primary ideals. By hypothesis, each primary ideal of R is a power of its radical. Thus, every ideal of R can be written as a product of prime ideals of R. Therefore, R is a ZPI-ring.

### Theorem 2.13

Let D be an integral domain which is not a field, then the following statements are equivalent:

• D is a Dedekind domain.

• D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology.

Proof

By Theorem 1.1 and 1.2, it is sufficient to prove (2) ⇒ (1). Suppose that D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology. Thus, by Theorem 2.2, every proper ideal I of D can be written as I = J1J2 ∩ … ∩ Jn, where J1, J2,…,Jn are ideals of D such that each of the D/Ji is indecomposable. By Proposition 2.11, each of the Ji is primary, and so every proper ideal of D can be written as an intersection of primary ideals. Since D is an almost Dedekind domain, each nonzero primary ideal of D is a power of its radical. Thus, every nonzero proper ideal I of D can be written as I = J1J2 ∩ … ∩ Jn, such that each of the Ji is a power of a maximal ideal, say Ji=miri for some ri ∈ ℕ. Hence, I=J1J2Jn=m1r1m2r2mnrn=m1r1m2r2mnrn. Therefore, D is a Dedekind domain.

### Theorem 2.14

Let D be an integral domain which is not a field, then the following statements are equivalent:

• D is an almost Dedekind domain and each nonzero element of D is contained in only finitely many maximal ideals of D.

• D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology.

Proof

By Theorem 2.13 and [4, Theorem 37.2].

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