### Article

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

**Published online** September 23, 2017

Copyright © Kyungpook Mathematical Journal.

### 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

It is well known that an integral domain

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

### 1. Introduction

Let

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])

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 ideal of D, the ring of quotients is a valuation ring.

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

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])

R is a ZPI-ring. R is a Noetherian ring such that for each maximal ideal of R, there are no ideals of R strictly between and .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(

### 2. Main Results

Recall that a ring _{1} × _{2}, where _{1} and _{2} are both nonzero rings. It is well known, and not difficult to prove, that a ring

### Lemma 2.1

The ideal I can be written as I =J _{1}∩J _{2}∩ … ∩J _{n}, where J _{1}, J _{2}, …,J _{n}are ideals in R such that each of the R /J _{i}is indecomposable. The ideal I can be written as I =K _{1}K _{2}…K =_{m}K _{1}∩K _{2}∩…∩K _{m}, where K _{1}, K _{2}, …,K _{m}are pairwise comaximal ideals in R such that each of the R /K _{i}is indecomposable. R /I has only finitely many idempotents.

**Proof**

(1) ⇒ (2) Let _{1}∩_{2}∩…∩_{n}_{1}, _{2},…,_{n}_{i}_{i}_{1} and _{2}, are contained in a maximal ideal of _{1}∩_{2}) is an idempotent in _{1} ∩_{2}), then _{1} and _{2} are idempotents in _{1} and _{2}, respectively. Now since _{1} and _{2} are indecomposable and _{1} ⊆ and _{2} ⊆ , we have _{1} ∩ _{2}) = 0+(_{1} ∩ _{2}) or _{1} ∩ _{2}) = 1+(_{1} ∩ _{2}). Hence, _{1} ∩ _{2}) is also indecomposable. Set _{3}_{n}_{1} ∩_{2} ∩ … ∩_{m}_{1}, _{2},…,_{m}_{i}_{1}_{2},…,_{m}_{1}_{2}…_{m}_{1} ∩ _{2} ∩ … ∩ _{m}_{1}_{2}…_{m}_{1} ∩ _{2} ∩ … ∩ _{m}_{1}, _{2},…,_{m}_{i}_{i}

(3) ⇒ (1) If

For a ring

A topological space

### Theorem 2.2

_{1} ∩ _{2} ∩ … ∩ _{n}_{1}_{2}_{n}_{i}

**Proof**

Let _{1}∩_{2}∩…∩_{n}_{1}, _{2},…,_{n}_{i}_{1}, _{2},…,_{n}_{1} = _{1}, _{2} = _{1}_{2},…, _{n}_{1}_{2}…_{n}_{i}_{i}_{+1} = _{i}_{+1}. Set _{i}_{i}_{i}_{+1} for each _{i}_{i}

Set _{n}_{i}_{∈ℕ{1,2,…,}_{n}_{}} for each _{1} ⊆ _{2} ⊆ _{3} ⊆ …, is an ascending chain of J-radical ideals of _{k}_{k}_{+1}. Therefore, _{k}_{+1} = ∩_{i}_{∈ℕ{1,2,…,}_{k}_{+1}} ⊆ _{k}_{i}_{∈ℕ{1,2,…,}_{k}_{}} ⊆ . Now for all _{k}_{+1} = _{k}_{+1} − 0 = (1 − _{i}_{k}_{+1} ∈ . Thus, _{k}_{+1} ∈ _{k}_{+1} ⊆ . Therefore, _{k}_{+1} ∈ and 1 − _{k}_{+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

An ideal

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

By Lemma 2.3, if

### Corollary 2.5

### Lemma 2.6

**Proof**

Let

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

### Remark 2.8

It is easily seen that if an ideal

### Lemma 2.9

^{2} =

**Proof**

Let ^{2}+^{2}−^{2}+^{2}−^{2} = ^{2} − ^{2} − ^{2} − ^{2} = (1−^{2} = (1−^{2}^{2} = (1−^{2} = (1−

### Lemma 2.10

_{α}_{α}_{∈Λ}_{α}_{∈Λ}_{α}

**Proof**

Let {_{α}_{α}_{∈Λ} be a family of prime ideals of _{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}_{α}_{∈Λ}_{α}

### Proposition 2.11

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 ^{2} =

Now let _{I}_{⊆}_{P}_{∈Spec(}_{R}_{)} is a zero-dimension ring, as above,

Now let _{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}

**Case 1**

If ∩_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}^{n}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}

**Case 2**

If ∩_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{1} of a ring _{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{1}∩_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}_{I}_{⊆}_{P}_{∈Spec(}_{R}_{)}

Now we can state the main results of this paper.

### Theorem 2.12

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 _{1}_{2}…_{m}_{1} ∩_{2} ∩ … ∩_{m}_{1}, _{2},…,_{m}_{i}_{i}

### Theorem 2.13

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 _{1} ∩ _{2} ∩ … ∩ _{n}_{1}, _{2},…,_{n}_{i}_{i}_{1} ∩ _{2} ∩ … ∩ _{n}_{i}_{i}

### Theorem 2.14

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].

- Anderson, FW, and Fuller, KR (1992). Rings and categories of modules. Graduate Texts in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag
- Gilmer, R (1962). Rings in which semi-primary ideals are primary. Pacifie J Math.
*12*, 1273-1276. - Gilmer, R (1964). Extension of results concerning rings in which semi-primary ideals. Duke Math J.
*31*, 73-78. - Gilmer, R (1972). Multiplicative ideal theory. Pure and Applied Mathematics. New York: Marcel Dekker
- Larsen, M, and McCarthy, P (1971). Multiplicative theory of ideals. Pure and Applied Mathematics. New York: Academic Press
- Loper, A (2006). Almost Dedekind domains which are not Dedekind. Multiplicative ideal theory in commutative algebra. A tribute to the work of Robert Gilmer, Brewer, W, Glaz, S, Heinzer, W, and Olberding, B, ed. New York: Springer, pp. 276-292
- Ohm, J, and Pendleton, RL (1968). Ring with Noetherian spectrum. Duke Math J.
*35*, 631-639.