### Article

Kyungpook Mathematical Journal 2020; 60(1): 53-69

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

### Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

David Earl Dobbs

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 379961320, USA

e-mail : dedobbs@comporium.net

**Received**: June 12, 2019; **Revised**: January 7, 2020; **Accepted**: January 10, 2020

Let (

**Keywords**: commutative ring, ring extension, minimal ring extension, inert extension, maximal ideal, minimal ﬁ,eld extension, quasi-local ring, integrality, pullback.

### 1. Introduction

All rings and algebras considered below are commutative and unital; all inclusions of rings, ring extensions and algebra/ring homomorphisms are unital. If _{P}_{AP}

Let _{M}_{M}_{P}_{P}

A minimal ring extension ^{2}). Now let ^{2}). Notice that in this situation, where the minimal ring extension _{M}_{M}

The above definition of an “inert extension” should not be confused with the use of this terminology in papers, such as [5], that follow the usage in a well-known paper of P. M. Cohn [4]. Remark 2.17 establishes that these two notions of an “inert extension” are logically independent.

Much is known about the possible existence of the various kinds of integral minimal ring extensions (that is, ramified, decomposed or inert). For instance, if

Existence results for inert extensions are not as widely available as for ramified or decomposed extensions. Indeed, some rings

However, some clarity (if not a complete pattern for the existence of inert extensions) was provided by the following result [11, Theorem 2.5 (b)]. If (^{*} such that ^{*} ∈ [^{*} ⊂ ^{*}-algebra (in the sense of [24]) and each such ^{*} are finite local rings with the same maximal ideal; it can be arranged that (one such) ^{*} is ^{*} shares with ^{*} ⊂ ^{*} ⊂ ^{*}-algebra and ^{*} ⊂

Recall that [11, Theorem 2.5 (b)] established that each finite local ring ^{*} (given

We begin by showing in Example 2.1 that there exists an integral (non-minimal) ring extension (

Example 2.8 complicates the focusing process by producing quasi-local rings (_{B/N} B

Further information about minimal ring extensions can be found in some of the above-mentioned references, as well as in [28] and [31]. Any unexplained material is standard, as in [20] and [26].

### 2. Results

Recall from [11, Theorem 2.5 (b)] that if (^{*}. This

The easiest kind of example illustrating this fact is provided by considering any nonzero quasi-local (finite, if you wish) ring (

The proof of Example 2.1 will use the following well known facts. Let

### Example 2.1

Let

**Proof of Example 2.1**

By definition, _{L}B

Next, since

It will suffice to prove that

Suppose the assertion fails. Pick some

The behavior of the data in Example 2.2 will be very different from that in Example 2.1, even though we will require that

### Example 2.2

Let ^{p}

**Proof of Example 2.2**

Since ^{p}^{p}^{p}^{p}^{p}^{p}^{p}^{p}^{1}) ≠ ^{p}^{2}) whenever _{1} and _{2} are distinct odd prime numbers.) In view of a classic irreducibility result in field theory (which holds even if ^{p}^{p}^{p}

Suppose, on the contrary, that such an element ^{p}_{1}, _{2} ∈ ^{p}^{p}_{1}_{2}_{1}_{2} and so _{1}, _{2} ∈ ^{p}_{1}) ∈ _{2}) ∈

Dividing by

In response to the surfeit of rings

### Example 2.3

Fix a positive integer _{1}, _{n}_{i}_{i}^{2}) ∈ ^{2}). Then ^{2} = 0 ≠ ^{2}) can be written additively as _{i}_{i}

**Proof of Example 2.3**

Since and

Next, as in the proof of Example 2.1, the assignment _{1} ⊂ _{2} of finite fields is a minimal field extension if and only if [_{2}:_{1}] is a prime number. It follows that if _{i}_{i}_{1}_{2} ≤ _{i}_{1} ≠ _{i}_{2} since ; that is, _{i}_{1}_{i}_{2}_{i}_{1} ≠ _{i}_{2}. Thus, the _{i}

Since the field extension

### Remark 2.4

In the context of Example 2.3, _{i}_{i}_{i}_{i}/M_{B/M} B_{i}_{L} B

Consider quasi-local rings (_{C}_{C}_{C}

### Example 2.5

Let

K ⊂L be a minimal field extension, and let (B,N ) be any valuation domain which is not a field but is of the formB =L +N . PutA :=K . Then (A,M ) ⊂ (B,N ) are quasi-local rings such thatM := 0 ⊂N , the induced extension of residue fieldsA/M ⊆B/N is algebraic,D :=K +N ∈ [A,B ] is such thatD ⊂B is an inert (minimal ring) extension, and the ring extensionA ⊂B is not integral.If 1 ≤

n ≤∞ is preassigned and the minimal field extensionK ⊂L is given, then there exists a valuation domainB as in (a) such that dim(B ) =n (and all the assertions in (a) hold).

**Proof of Example 2.5**

Given 1 ≤

It remains only to show that the ring extension

### Proposition 2.6

Let (A,M) ⊂ (B,N) be quasi-local rings such that M ⊆ N, the induced extension of residue fields A/M ⊆ B/N is algebraic, and dim(B) = 0. Then the ring extension A ⊂ B is integral.

**Proof**

Using the canonical isomorphism (_{B/N} B

### Remark 2.7

By Example 2.5 (b), the conclusion of Proposition 2.6 would fail if we delete the hypothesis that dim(

B ) = 0. (In detail, the proof of Example 2.5 shows that (A,M ) = (K , 0) ⊂ (L +N,N ) = (B,N ) are quasi-local rings such thatM = 0 ⊆N ; that the extensionA/M ⊆B/N identifies with the minimal, hence algebraic, field extensionK ⊂L ; and that the extensionA ⊂B is not integral. Of course, dim(B ) ≠ 0 sinceB is a quasi-local integral domain whose maximal idealN is nonzero, the point being thatB is not a field.)For an example of data satisfying the hypothesis of Proposition 2.6, one could take

B as any finite (quasi-)local ring having a proper subringA . For such data,B may or may not be a field: consider taking (A,B ) to be ( ) or ( ).

By tweaking the data from Example 2.5, we next present data that behave very differently, in the sense that a ring

### Example 2.8

Let (_{2ℤ}, with

**Proof of Example 2.8**

As in the proofs of Examples 2.1 and 2.5, we get that (

It remains only to show that

It is surely apparent from several of the above proofs that pullback constructions are relevant to a more general study of inert extensions. Parts (a) and (b) of Proposition 2.9 are a couple of straightforward results in this same vein. Proposition 2.9(c) is a related result that figures in the proof of our main result. Proposition 2.9 (c) is stated at a level of generality that, in our opinion, should be recorded. Although we did not pause to note the elementary result Proposition 2.9 (a) (iii) earlier, it could have been used to shorten the next-to-last paragraph of the proof of Example 2.5 and the end of the proof of Example 2.8. Indeed, if Proposition 2.9 (a) (iii) had been available there, we would have not needed to show first that the ring extension

### Proposition 2.9

Let I be a common ideal of rings A ⊆B. Then: A is (isomorphic to )the pullback A/I ×_{B/I}B.A ⊂B is a minimal ring extension if and only if A/I ⊂B/I is a minimal ring extension. A ⊆B is an integral ring extension if and only if A/I ⊆B/I is an integral ring extension.

Let A ⊆B be quasi-local rings with the same maximal ideal M. Then the following five conditions are equivalent: The ring extension A ⊂B is inert; The ring extension A ⊂B is integral and inert; A ⊂B is an integral minimal ring extension; A ⊂B is a minimal ring extension; A/M ⊂B/M is a minimal field extension.

Let (B,M )be a quasi-local ring, let π :B →B/M be the canonical surjection, let k be a proper subfield of B/M, and let A :=k × :_{B/M}B. Let jA →B be the injective ring homomorphism and let p :A →k be the surjective ring homomorphism that are canonically associated with the pullback definition of A. (So, if we view A ⊆B, then j is the inclusion map and p =π | )_{A}.Put ℳ :=j ^{−1}(M ). Then: A is a quasi-local ring with maximal ideal ℳ,and j (A )is a quasi-local ring with maximal ideal M. j (A ) ⊂B is an inert extension (necessarily with crucial maximal ideal M )if and only if k ⊂B/M is a minimal field extension.

**Proof**

(a) Let ^{−1}(

(ii) As in the proof of Example 2.1, the assignment

(iii) One could simply apply [17, Corollary 1.5 (5)] to the pullback description from (i). Alternatively, the concrete nature of this pullback allows for the following straightforward calculational arguments. For the “only if” assertion, if

(b) (1) ⇒ (2): By definition, every inert (minimal) ring extension is integral.

(2) ⇒ (3) ⇒ (4): Trivial.

(4) ⇒ (1): As

(1) ⇒ (5): This follows from the definition of an inert extension.

(5) ⇒ (1): Assume (5). Then

(c) (i) By applying [17, Theorem 1.4] to the pullback that defined ^{−1}(

(ii) By inspecting commutative diagrams, it is easy to see that ker(^{−1}(_{j}_{(}_{A}_{)}) =

We next present our main result.

### Theorem 2.10

_{B/N} B.

**Proof**

Consider the ring _{B/N} B

It remains only to prove that if some

The proof is complete.

We next isolate two special cases of Theorem 2.10

### Corollary 2.11

_{B/N} B.

**Proof**

In view of Theorem 2.10, it suffices to show that

### Corollary 2.12

_{B/N} B.

**Proof**

As any ring extension involving finite rings is integral, an application of Corollary 2.11 completes the proof.

It is natural to ask if the converse of Theorem 2.10 is valid. Despite expectations that may have been raised by [11, Theorem 2.5 (b)], Example 2.13 provides a negative answer to this question.

### Example 2.13

Let 1 ≤

**Proof of Example 2.13**

It is well known that there exists a valuation domain (

### Remark 2.14

In contrast to the relationship between Example 2.5 and Proposition 2.6, it should be noted that the case where dim(

Recall that [11, Theorem 2.5(b)] gave a chain (^{*}^{*} ⊂ ^{*} is uniquely determined by these conditions. This naturally leads to the question of whether the conditions in Theorem 2.10 force

### Example 2.15

There exist finite (quasi-) local rings (

**Proof of Example 2.15**

Note that ^{2}) ∈ ^{2} = 0 ≠ ^{2} = 0, it is clear that

### Remark 2.16

It may be of interest to illustrate Theorem 2.10 by constructing data that is somewhat in the spirit of Example 2.15 but also satisfies ^{2}. Then ^{2}, the extension of residue fields ^{2}(^{2} + ^{3} = ^{2} ⊂

In addition to the meaning given above (and in the already-cited references) to the terminology of an “inert extension,” several papers in the literature ascribe quite a different meaning to this terminology. As this other usage derives from a paper of P. M. Cohn [4], it will be referred to here as a “C-inert extension” to avoid confusion. By definition, a ring extension

### Remark 2.17

Although the two above-mentioned usages of “inert extension” are each motivated by factorization results in classical algebraic number theory, neither of these usages implies the other. To see this, note first that if ^{2} +^{2}] ⊂ ℝ[^{2} + ^{2}]). On the other hand, if

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