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
Keywords: integral domain, survival pair, survival extension, lying-over, prime ideal, integral, subring, Krull dimension, ring of algebraic integers, characteristic.
All rings considered in this note are commutative with identity. Any inclusion of rings of the form
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 : 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
We begin with a result of some independent interest.
The parenthetical assertion holds since (
We can now sharpen the above-mentioned part of [9, Corollary 2.3] for domains of characteristic 0.
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 (
We next sharpen the above-mentioned part of [9, Corollary 2.3] for domains of positive characteristic.
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.
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].
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].
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. Let pbe any prime number, and consider R:= ℤ pℤ. Any subring of Ris 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  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 than pℤ does not survive in R.
pbe any prime number. Put F:= . We will show that the “LO-pair” condition in [9, Corollary 2.3] is not redundant in characteristic pby finding an example of a GD-pair ( F,R) which is not an LO-pair. Let Xbe a transcendental element over F, and consider R:= F[ X] XF[ X]. It is known that if A∈ [ 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 extension F[ X] ⊂ Rdoes not have the survival property since ( X+ 1) F[ X] does not survive in R.
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