Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

Jarboui, Noômen; Al-Kuleab, Naseam; Almallah, Omar

doi:https://doi.org/10.1155/2020/7416893

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Volume 2020 | Article ID 7416893 | https://doi.org/10.1155/2020/7416893

Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

Noômen Jarboui,¹Naseam Al-Kuleab,¹and Omar Almallah²

Academic Editor: Elena Guardo

Received30 Jun 2020

Accepted03 Nov 2020

Published12 Nov 2020

Abstract

The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

1. Introduction

All rings and algebras considered in this paper are assumed to be commutative with the identity element; all subrings, ring extensions, algebras, and ring (resp., algebra) homomorphisms are assumed unital. If is a ring extension, it is convenient to let denote the set of intermediate rings (that is, the set of rings such that ). We shall call a ring in an -overring of . Such a ring is said to be a proper -overring of if . When is the total quotient ring of , then each ring is called an overring of . Such a ring is termed a proper overring of if . If is a ring-theoretic property and is a ring extension, then is said to be a -pair if each -overring of has a property (see [1], p. 34). In 1992, Gilmer and Heinzer have studied Artinian pairs of rings (cf. [1]). Recall that a ring is called Artinian if satisfies the descending chain condition for ideals. Examples of Artinian rings are finite dimensional algebras over a field (recall that an algebra over a field is said to be finite dimensional or infinite dimensional according to whether the -vector space is finite dimensional or infinite dimensional). Artinian rings, and especially local Artinian rings, play an important role in the algebraic geometry, for example, in deformation theory. It is worth noticing that one of the most important trends in commutative ring theory is the study of the influence of given systems of intermediate rings of a ring extension on the structure of the extension itself. Examples of such systems are the family of all -overrings, that of all proper -overrings, etc. (see [1–17]). For a ring-theoretic property and a ring extension , let denote the family of -overrings of such that does not satisfy . Recently, many authors have investigated the behaviour of ring extensions for which . In this case, the ring is called a maximal non- subring of . These ring extensions have been studied for various properties such as Noetherian, ACCP, Jaffard, universally catenarian, local, valuation, pseudovaluation, integrally closed, and Prüfer (cf. [3–5, 9–15, 17, 18]). We are interested in this paper in the property Artinian and the family of cardinality 1 or 2. Our work is motivated on the one hand by [5], in which the authors have studied maximal non-Noetherian subrings of a domain, and on the other hand by the abovementioned work of Gilmer and Heinzer concerning Artinian pairs (cf. [1]) and also by the increasing interest in ring extensions with many intermediate rings as explained above. In Section 2, we study ring extensions with only one non-Artinian intermediate ring. We show in Theorem 1 that there exists a unique intermediate ring between and such that is not Artinian if and only if is a maximal non-Artinian subring of if and only if is a closed minimal extension and is Artinian. As a consequence, if is an integral domain, then is a maximal non-Artinian subring of if and only if is a rank one valuation domain with quotient field (see Corollary 1). In Section 3, we study ring extensions having exactly two non-Artinian intermediate rings. We give full characterizations of these extensions in Theorems 2 and 3.

Let be a ring extension. Throughout this paper, denotes the integral closure of in and denotes the integral closure of (in its total quotient ring). We use “” for inclusion and “” for strict inclusion. Any undefined notation or terminology is standard, as in [19, 20].

2. Ring Extensions with Only One Non-Artinian Intermediate Ring

We start with the following result which is an easy consequence of [1] (Theorem 2) or [21] (Theorem 3.8).

Proposition 1. Let be a ring extension, and suppose that there is at least one -overring of which is not Artinian. Assume, moreover, that the class of non-Artinian -overrings of is finite. Then, .
We start our investigations by recalling some results about minimal ring extensions and normal pairs of rings. A ring extension is said to be minimal if is a proper subring of and . If is a minimal extension, then either , in which case, is called a closed minimal extension, or , in which case is called a minimal integral extension (see [22]). If is a ring extension, then is called a normal pair if was each -overring of . The concept of a normal pair was introduced in case is a (integral) domain by Davis [23]. The most natural example of a normal pair arises when is an arbitrary Prüfer domain and is its quotient field (cf. [23] (Theorem 1) or [19] (Theorems 23.4(1) and 26.1(1))). In [24], Ayache and Jaballah have pursued the study of normal pairs of integral domains. So, several characterizations of such pairs have been obtained. In [25–27], the authors have studied normal pairs of rings with zero divisors, so many results are generalized from the domain-theoretic case to arbitrary rings.
Recalling from [28] that given rings and an element , we say that is primitive over in case is a root of a polynomial with unit content, that is, the coefficients of generate the unit ideal of . If each element of is primitive over , then is said to be a -extension. Following [20] (p. 28), we let INC denote the incomparability property of ring extensions (so a ring extension satisfies INC if and only if distinct comparable prime ideals of must contract to distinct prime ideals of ). As in [29], if is a ring extension, we say that is an INC pair if satisfies INC for each -overring of . It was proved in [30] (Theorem) that is a -extension if and only if is an INC pair. The authors in [25] (Theorem 1) ensure that is a normal pair if and only if is a -extension and is integrally closed in .

The next theorem characterizes ring extensions with only one non-Artinian intermediate ring.

Theorem 1. Let be a ring extension. Then, the following statements are equivalent:(1)There exists a unique intermediate ring between and such that is not Artinian(2) is a maximal non-Artinian subring of (3) is a (closed) minimal extension and is Artinian

Proof. (1)(2). Proposition 1 asserts that is not Artinian, and as, by assumption, there exists a unique intermediate ring between and which is not Artinian, it follows that is a maximal non-Artinian subring of . (2)(3). First, we note that is a -extension or equivalently is an INC pair. Indeed, if is a proper -overring of , then is zero dimensional (since it is Artinian), so clearly the ring extension satisfies incomparability. Now, we claim that is integrally closed in . Let be an integral over and suppose that . Then, is a proper -overring of . Hence, is an Artinian ring. As is an integral extension and is zero dimensional, we infer that so too is . It is also evident that is Noetherian, and hence is Artinian, which is a contradiction. We conclude using [25] (Theorem 1), see also the last comments in our introduction, that is a normal pair. We will demonstrate that is a minimal ring extension. To this end, suppose that is a proper -overring of , then is a zero dimensional pair. Thus, the authors in [31] (Corollary 4.2) ensure that is an integral over . But, by what we have already observed, must be a normal pair; hence, , and we are done. (3)(1). It is enough to show that the ring is not Artinian. To this end, assume the contrary. Then, as is Artinian, would be an integral extension by virtue of [31] (Corollary 4.2), which is a contradiction. This completes the proof.

Next, we treat the particular case, where is an integral domain.

Corollary 1. Let be an extension of integral domains. Then, the following statements are equivalent:(1) is a maximal non-Artinian subring of (2) is a rank one valuation domain with quotient field

Proof. (1)(2). As is an Artinian integral domain, then is a field. According to Theorem 1, the ring extension is a closed minimal extension. Thus, is the quotient field of and is a rank one valuation domain (2)(1). Trivial

3. Ring Extensions with Exactly Two Non-Artinian Intermediate Rings

We start with the following result.

Proposition 2. Let be a ring extension having exactly two non-Artinian intermediate rings. Then, either is a minimal extension and is not Artinian or there is an intermediate ring such that and are minimal extensions, is integrally closed in , is Artinian, and is not Artinian.

Proof. According to Proposition 1, is not Artinian. Let be the second non-Artinian ring such that . Notice that is a minimal ring extension. Indeed, assume the contrary and let be a ring such that . As each -overring of properly contained in is Artinian, then would be Artinian by virtue of [1] (Theorem 2), a contradiction. If , then is a minimal extension with both and non-Artinian. Suppose now that . As the ring extension has exactly two non-Artinian intermediate rings, then is Artinian. Since is not Artinian and each proper -overring of is Artinian, then would be the maximal non-Artinian subring of . Thus, Theorem 1 guarantees that is a closed minimal extension.

In the next theorem, we identify ring extensions with exactly two non-Artinian intermediate rings in case is integrally closed in .

Theorem 2. Let be a ring extension such that is integrally closed in . Then, the following statements are equivalent:(1)There are exactly two non-Artinian intermediate rings between and (2)Either is a (closed) minimal extension with non-Artinian or is a chain of length 2 and is Artinian

Proof. (1)(2). If is a minimal extension, then we are done. Hence, suppose now that is not a minimal extension. Then, according to Proposition 2, there exists an intermediate ring such that and are minimal extensions, is Artinian, and is not Artinian. In view of [25] (Theorems 1 and 2), is a normal pair. Now, we claim that . Indeed, suppose that . Then, is an Artinian pair. Hence, is integral over ([31], Corollary 4.2). But, is integrally closed in since is a normal pair. Thus, , a contradiction. (2)(1). If is a minimal extension with non-Artinian, then according to [1] (Theorem 2), is not Artinian. Hence, we are done. Assume now that is a chain of length 2 and is Artinian. Then, , where and are (closed) minimal extensions. The ring is not Artinian since otherwise would be Artinian pair and so would be an integral over according to [31] (Corollary 4.2), which is impossible since is integrally closed in . Moreover, is not Artinian since otherwise would be Artinian by virtue of [1] (Theorem 2). Therefore, there are exactly two non-Artinian intermediate rings between and , namely, and . This completes the proof.

In the following theorem, we determine all ring extensions with exactly two non-Artinian intermediate rings in case is not integrally closed in . But, first some facts about minimal ring extensions are recalled. According to [22] (Théorème 1(i) and Lemme 1.3), if is a minimal extension and is not a field, then there exists a unique maximal ideal of called the crucial maximal ideal of such that the canonical injective ring homomorphism can be viewed as a minimal ring extension, while the canonical ring homomorphism is an isomorphism for all prime ideals of , except . If in addition is an integral extension, then is precisely the conductor (cf. [22], Théorème 1(ii)).

Theorem 3. Let be a ring extension such that is not integrally closed in . Then, the following statements are equivalent:(1)There are exactly two non-Artinian intermediate rings between and (2)Either is a minimal integral extension with non-Artinian, or is a chain of length 2 such that is Artinian and is not Artinian, or consists of two chains of length 2 such that is Artinian and is not Artinian

Proof. (1)(2). If is a minimal extension, then it must be integral since is not integrally closed in . Moreover, as there are exactly two non-Artinian intermediate rings between and , then and should be non-Artinian. Now, assume that is not a minimal extension. Then, according to Proposition 2, there exists an intermediate ring such that and are minimal extensions, is Artinian, and is not Artinian. Moreover, is a closed minimal extension. It follows that the minimal ring extension is an integral. In this case, and hence . Consider the set: Claim 1. has a maximal element Indeed, it is obvious that is nonempty since . The set equipped with the inclusion relation is a partially ordered set. Let now be a totally ordered subfamily of , and let . One can easily check that is an intermediate ring between and , and . Thus, . It follows, by virtue of Zorn’s lemma, that has a maximal element. Claim 2. has a greatest element According to claim 1, has a maximal element, say . We will show that is the greatest element of . If , then clearly is the greatest element of . Assume now that and then contains properly. It is worth noticing that if , then and are incomparable under inclusion. So, . Hence, is Artinian. Clearly, is a maximal non-Artinian subring of . It follows that is a closed minimal extension, by Theorem 1. In particular, is a closed minimal extension. It is not difficult to check that is a minimal integral extension. Let . We need to prove that . If , then by maximality of , we get . Assume now that and . Let . As is a proper -overring of and is minimal, then necessarily . Let be the crucial maximal ideal of the minimal extension . It follows from [32] (Lemma 2.3) that . As is integrally closed in and in , then a fortiori . Hence, , which is the desired contradiction. We deduce that and so is the greatest element of . If , then is a chain of length 2. More precisely, . Indeed, let . If , then . So . If , then . So or . Now, if , we claim that . Indeed, let . If , then . So or because is minimal as noted above. If , then contains and so or , which is the desired conclusion. (2)(1). If is a minimal extension and is not Artinian, then we are done, by Proposition 1. Suppose now that is a chain of length 2, is Artinian, and is not Artinian. We claim that . Indeed, as is not integrally closed in , then . Now, assume that . As is a chain of length 2, then there exists a ring such that . The ring cannot be Artinian, since otherwise would be a maximal non-Artinian subring of and so would be integrally closed in by virtue of Theorem 1. Thus, , which is absurd. It follows that as claimed. The ring cannot be Artinian, by Theorem 1. Therefore, there are exactly two non-Artinian intermediate rings between and , namely, and . Now, assume that consists of two chains of length 2, is Artinian and is not Artinian. Then, there exist two incomparable rings and distinct from and from such that . First, we handle the case where . Suppose that for some is not Artinian, then would be a maximal non-Artinian subring of . So, would be integrally closed in , by Theorem 1, which is impossible since is integral over and . Thus, and are Artinian. It follows that is a maximal non-Artinian subring of . Thus, Theorem 1 ensures that is integrally closed in , a contradiction since is integral over and . Thus, we conclude that . Without loss of generality, we can suppose that . The ring cannot be Artinian since otherwise by [31] (Corollary 4.2), we get , which is absurd. The ring is Artinian. Indeed, assuming the contrary, then would be a maximal non-Artinian subring of and hence would be a closed minimal extension according to Theorem 1. Thus, is a minimal integral extension, for otherwise would be integrally closed in , which contradicts the assumption made on . Hence, , a contradiction with the fact that and are incomparable. Therefore, there are exactly two non-Artinian intermediate rings between and , namely, and . The proof is complete.

The following corollary treats the particular case, where is an integral domain. It is worth noticing that an important step, toward the classification of minimal extensions of integral domains, was taken by Sato–Sugatani–Yoshida, who showed in [33] (page 1738, lines 8–13) that if is a minimal extension such that is not a field, then is an overring of .

Corollary 2. Let be an extension of integral domains. Then, the following statements are equivalent:(1)There are exactly two non-Artinian intermediate rings between and (2)Either is a minimal extension and is not a field, or is a rank two valuation domain with quotient field , or is a minimal extension and is a rank one valuation domain with quotient field

Proof. (1)(2). If is a minimal extension, then we are done. Thus, suppose that is not a minimal extension. It follows from Theorems 2 and 3 that is a field and is not a field. If is integrally closed in , then is a chain of length 2. According to [33] (page 1738, lines 8–13), . Thus, is a rank two valuation domain with quotient field . If is not integrally closed in , then Theorem 3 and [33] (page 1738, lines 8–13) guarantee that is a chain of length 2. Hence, is a minimal extension and is a rank one valuation domain with quotient field . (2)(1). If is a minimal extension and is not a field, then cannot be a field (see [22], Théorème 1). If is a rank two valuation domain with quotient field , then , where is a rank one valuation overring of . Finally, if is a minimal extension and is a rank one valuation domain with quotient field , then the authors in [34] (Theorem 2.4) ensure that . Therefore, in all cases, there are exactly two non-Artinian intermediate rings between and . This completes the proof.

We close the paper with the following example. The authors would like to thank Professor Gabriel Picavet for providing them this example.

Example 1. This example provides a ring extension such that consists of two chains of length 2, namely, and , such that and are Artinian, whereas and are not Artinian.
Let be a discrete valuation domain with quotient field , so that is a minimal closed extension, and let be a minimal field extension (and then minimal integral). Set , and . Clearly, and are Artinian since they are products of two fields. Moreover, one can easily check that , and . In view of [35] (Proposition 4.7), we get that and are minimal integral extensions, while and are minimal closed extensions. This leads to . Now, it is not difficult to check that (resp., ) is the crucial maximal ideal of (resp., ). Since , crosswise exchange lemma (cf. [36], Lemma 2.7) asserts that . In particular, consists of two chains of length 2: and . The ring (resp., ) is not Artinian because (resp., ) is a prime nonmaximal ideal of (resp., ).

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the Deanship of Scientific Research at King Faisal University for the financial support under the annual research project (Grant no. 180087).

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Copyright © 2020 Noômen Jarboui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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