### Article

Kyungpook Mathematical Journal 2017; 57(2): 187-191

**Published online** June 23, 2017

Copyright © Kyungpook Mathematical Journal.

### Using Survival Pairs to Characterize Rings of Algebraic Integers

David Earl Dobbs

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.

**Received**: May 22, 2016; **Accepted**: May 4, 2017

Let

**Keywords**: integral domain, survival pair, survival extension, lying-over, prime ideal, integral, subring, Krull dimension, ring of algebraic integers, characteristic.

### 1. Introduction

All rings considered in this note are commutative with identity. Any inclusion of rings of the form _{1} ⊆ _{2} will be interpreted to mean that _{1} is a unital subring of _{2}. As usual, Spec(

It is well known that GU ⇒ LO ⇒ survival (for extensions) and neither of these implications is reversible. Thus, GU-pair ⇒ LO-pair ⇒ survival-pair. Though it is not at all obvious, both of these implications are reversible. The fact that every LO-pair is a GU-pair was shown in [8, Corollary 3.2]. The proof that every survival pair is an LO-pair was given as part of the proofs of [5, Lemma 2.1] and [5, Theorem 2.2]. It may be fair to say that GD-pairs have been studied less than survival pairs (if one takes into account that survival pairs are the same as LO-pairs). However, both of these types of pairs figure prominently in our starting point, which is the principal result of [10]: if

The answer is “Yes.” More precisely, our main result (Theorem 2.2) is that if one considers only domains of characteristic 0, the above condition on GD-pairs is redundant. In other words, Theorem 2.2 establishes that if

As usual, if

### 2. Results

We begin with a result of some independent interest.

### Lemma 2.1

**Proof**

The parenthetical assertion holds since (_{A}^{n}^{n}

We can now sharpen the above-mentioned part of [9, Corollary 2.3] for domains of characteristic 0.

### Theorem 2.2

**Proof**

By the Lying-over Theorem ([12, Theorem 11.5], [13, Theorem 44]), every integral ring extension satisfies LO and, hence, satisfies the survival property. This fact establishes the “only if” assertion. For the converse, suppose that (

Consider _{1} := _{1}. Let _{1} is a Noetherian ring. Moreover, _{1} is integrally closed in _{1}_{1}. Thus, _{1} ⊆

We next sharpen the above-mentioned part of [9, Corollary 2.3] for domains of positive characteristic.

### Proposition 2.3

Either R =K is an algebraic field extension of A or precisely one valuation domain of K does not contain R; (

A,R )is a survival pair.

**Proof**

By [9, Corollary 2.3], it suffices to prove that if (

Combining Theorem 2.2 and Proposition 2.3, we can now give the following characteristic-free formulation of a sharpening of part of [9, Corollary 2.3].

### Corollary 2.4

The next remark gives two examples showing that Theorem 2.2 and Proposition 2.3 found the only redundancies in the above-mentioned part of [9, Corollary 2.3].

### Remark 2.5

We will show that the “LO-pair” condition in [9, Corollary 2.3] is not redundant in characteristic 0 by finding an example of a GD-pair (ℤ

, R ) which is not an LO-pair. Letp be any prime number, and considerR := ℤ_{p}_{ℤ}. Any subring ofR is an overring of ℤ and, hence by [12, Theorem 26.1 (1)], must be a Prüfer domain. Thus, as mentioned in the Introduction, it follows from [14] that (ℤ, R ) is a flat pair. Since any flat ring extension satisfies GD (cf. [13, Exercise 37, page 44]), (ℤ, R ) is a GD-pair. (This could also be established by showing that each ring in [ℤ, R ] is one-dimensional.) However, (ℤ, R ) is not an LO-pair since any nonzero prime ideal of ℤ other thanp ℤ does not survive inR .Let

p be any prime number. PutF := . We will show that the “LO-pair” condition in [9, Corollary 2.3] is not redundant in characteristicp by finding an example of a GD-pair (F,R ) which is not an LO-pair. LetX be a transcendental element overF , and considerR :=F [X ]_{XF}_{[}_{X}_{]}. It is known that ifA ∈ [F,R ], then dim(A ) ≤ 1 (cf. [12, Theorem 30.11 (a)]). Thus, (F,R ) is a GD-pair. However, (F,R ) is not an LO-pair. Indeed, the extensionF [X ] ⊂R does not have the survival property since (X + 1)F [X ] does not survive inR .

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