Houston Journal of Mathematics. c 2002 University of Houston Volume 28, No. 1, 2002

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1 Houston Journal of Mathematics c 2002 University of Houston Volume 28, No. 1, 2002 ON PSEUDO-VALUATION DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS R. DOUGLAS CHATHAM AND DAVID E. DOBBS Communicated by Johnny A. Johnson Abstract. It is proved that if R T are going-down domains such that Spec(R) = Spec(T ) as sets (for instance, a proper field extension) and M denotes the common maximal ideal of R and T, then each ring between R and T is a going-down domain if and only if the transcendence degree of T/M over R/M is at most 1. As a consequence, transcendence degree is used to characterize the pseudo-valuation domains all of whose overrings are going-down domains. 1. Introduction All rings considered below are (commutative integral) domains. Recall from [4] and [9] that a domain R is called a going-down domain if R T satisfies the going-down property for each domain T containing R. The most familiar examples of going-down domains are Prüfer domains and domains of (Krull) dimension 1. The main purpose of this note is to generalize ([4], Corollary 4.4 (ii), (iii)). In that result, one began with a nonalgebraic field extension k L and defined a sequence of rings R n, n 1, by R 1 := [[k, L)) := k + XL[[X]] and for n 1, R n+1 := [[R n, K n )), where K n denotes the quotient field of R n. It is easy to see that if n 1, then R n is an n-dimensional going-down domain. Moreover, it was shown in ([4], Corollary 4.4 (ii), (iii)) that if n 1, then each overring of R n is a going-down domain if and only if td k (L), the transcendence degree of L over k, is 1. We next explain how the key to our generalization of the above result from [4] is provided by the concept of pseudo-valuation domain (P V D) Mathematics Subject Classification. Primary 13G05, 13B24, 13F05; Secondary 13C15, 12F99, 13F30, 13B21. Key words and phrases. Going-down domain, pseudo-valuation domain, overring, transcendence degree, prime ideal, pullback. 13

2 14 R. DOUGLAS CHATHAM AND DAVID E. DOBBS Recall from [11] that a domain R is called a P V D if there is a (uniquely determined) valuation overring V of R such that Spec(R) = Spec(V ) as sets; in this case, V is called the canonically associated valuation overring of R. It was shown in ([1], Proposition 2.6) that the P V Ds may be characterized as the pullbacks of the form R = V V/M k, where (V, M) is a valuation domain and k is a subfield of V/M; in this case, V is necessarily the canonically associated valuation overring of R. Each P V D is a going-down domain (combine ([6], p. 560) and ([5], Proposition 2.1)), but the converse is false. With this background on P V Ds in hand, let us return to the rings R n described above. By ([6], Proposition 4.9 (i)), R n is a P V D for each n 1. Moreover, it is straightforward to verify that the canonical pullback description of R n is essentially R n = V n L k. In other words, for all n 1, if (V n, M n ) denotes the canonically associated valuation overring of R n, then one has canonical compatible isomorphisms, R n /M n = k and V n /M n = L. This observation motivates our titular result, Corollary 2.6 (a): if a P V D has the canonical pullback description R = V L k (where (V, M) is the canonically associated valuation overring of R, k = R/M and L = V/M), then each overring of R is a going-down domain if and only if td k (L) 1. Corollary 2.6 (a) is the desired generalization of ([4], Corollary 4.4 (ii), (iii)). Finally, we note that Corollary 2.6 (b) and Example 2.7 (b) make contact with the result (cf. ([12], Proposition 2.7)) that if R = V L k is a P V D as above, then each overring of R is a P V D if and only if L is algebraic over k, that is, if and only if td k (L) = 0. We proceed somewhat more generally than suggested above. Our main result, Theorem 2.5, concerns an extension of distinct domains, R T, such that Spec(R) = Spec(T ) as sets. Under this condition, it is known that R is quasilocal ([1], Lemma 3.2), say with maximal ideal M, so that R = T L k, where k := R/M and L := T/M; and that R is a going-down domain if and only if T is a going-down domain ([1], Proposition B.2). Theorem 2.5 establishes, for this context in which R is a going-down domain, that each ring between R and T is a going-down domain if and only if td k (L) 1. The proofs of Lemma 2.4 and Theorem 2.5 use standard results on pullbacks, as in ([10]), and a pullback-theoretic characterization of going-down domains ([7]). It is convenient to use the following standard notation and terminology. If A is a ring (domain), then Spec(A) denotes the set of prime ideals of A and dim(a) denotes the Krull dimension of A; and, by an overring of A, we mean a ring contained between A and the quotient field of A. Finally, if R T are domains, then td R (T ) denotes the transcendence degree of (the quotient field of) T over (the quotient field of) R.

3 ON PVDS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS Results It is well known (cf. ([13], Theorems 16 and 44)) that if k L is a field extension, then each ring between k and L is a field if and only if L is algebraic over k, that is, if and only if td k (L) = 0. We begin by showing that the td k (L) 1 analogue naturally involves going-down domains. Theorem 2.1. If k L are fields, then the following conditions are equivalent: (1) Each ring contained between k and L is a going-down domain; (2) A B satisfies the going-down property, for all rings k A B L; (3) dim(a) 1 for each ring A contained between k and L; (4) td k (L) 1. Proof. (3) (1): Each domain of Krull dimension at most 1 is a going-down domain. (1) (2): Apply the definition of going-down domain that was recalled in the second sentence of the Introduction. (2) (4): Deny. Then there exist X, Y L such that X and Y are algebraically independent over k. It follows from (2) that if A := k[x, Y ], then A B satisfies the going-down property for each overring B of A (since L contains k(x, Y ), the quotient field of A). However, a fundamental result ([9], Theorem 1) shows that a domain D is a going-down domain if (and only if) D E satisfies the going-down property for each overring E of D. Therefore, A is a going-down domain. As A is Noetherian (by the Hilbert Basis Theorem), it follows from ([4], Corollary 2.3) that dim(a) 1, a contradiction, for dim(a) = 2 (cf. ([14], Proposition 13, p. III-19), ([13], Theorem 39)). (4) (3): Deny. Then some ring A contained between k and L has distinct nonzero prime ideals P 1 P 2. Choose 0 u P 1 and v P 2 \P 1 ; and consider B := k[u, v]. Evidently, P 1 B P 2 B are distinct nonzero prime ideals of B, and so dim(b) 2. However, since B is algebra-finite over k, a corollary of the Noether Normalization Lemma ([14], Proposition 14, p. III-22) yields that dim(b) = td k (B). As 2 td k (B) td k (L) 1 by (4), we have the desired contradiction. We next relax the condition that the top ring in the extension be a field. Corollary 2.2. If a domain T contains a field k, then the following conditions are equivalent: (1) Each ring contained between k and T is a going-down domain; (2) Each ring contained between k and the quotient field of T is a going-down

4 16 R. DOUGLAS CHATHAM AND DAVID E. DOBBS domain; (3) td k (T ) 1. Proof. (2) (3) by Theorem 2.1; and (2) (1) trivially. It suffices to prove that (1) (3). Deny. Then the quotient field of T contains two elements, say X 0 and Y 0, that are algebraically independent over k. Write X 0 = t 1 /t 2 and Y 0 = t 3 /t 4, for suitable nonzero elements t 1, t 2, t 3, t 4 T. Consider A := k[t 1, t 2, t 3, t 4 ] T. As X 0 and Y 0 belong to F, the quotient field of A, we see that td k (F ) 2. Since F = k(t 1, t 2, t 3, t 4 ), it follows that {t 1, t 2, t 3, t 4 } contains a transcendence basis of F over k. Therefore, A contains two elements, say X and Y, that are algebraically independent over k. It follows from (1) that k[x, Y ] is a going-down domain, a contradiction (as in the proof of Theorem 2.1). We turn next to a case in which the bottom ring is not a field. First, it is convenient to recall that a domain R is said to be treed if no maximal ideal of R contains a pair of prime ideals that are incomparable (under inclusion). Each going-down domain is treed ([4], Theorem 2.2), but the converse is false. Proposition 2.3. Let R T be domains such that R is not a field and each ring contained between R and T is treed (for instance, a going-down domain). Then T is algebraic over R. Proof. Deny. Then there exists X T such that X is transcendental over R. By hypothesis, A := R[X] is a treed domain. However, if M denotes a nonzero maximal ideal of R (such M exists since R is not a field), then M + XA is a maximal ideal of A that contains the incomparable prime ideals MA and XA, contradicting that A is treed. For an extensive study of themes suggested by (2.1)-(2.3), see ([3]). The remainder of this note addresses ring extensions generalizing the inclusion of a P V D in its canonically associated valuation overring. Recall from ([1], Propositions B.2 and 2.2) that if distinct domains R T satisfy Spec(R) = Spec(T ), then R is a going-down domain (resp., P V D) if and only if T is a going-down domain (resp., P V D). Lemma 2.4. (a) Let R T be distinct going-down domains such that Spec(R) = Spec(T ) as sets. Let M denote the maximal ideal of R (and of T ). If S is a ring contained between R and T, then M Spec(S) and S M is a going-down domain. (b) Let (R, M) be a P V D, with canonically associated valuation overring V. If S is a ring contained between R and V, then M Spec(S) and S M is a P V D, with canonically associated valuation overring V.

5 ON PVDS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS 17 Proof. (a) Observe that M = M S Spec(S). Therefore, S = T L D, where D := S/M and L := T/M. According to ([10], Proposition 1.9), localizing each vertex of the underlying pullback diagram at the multiplicatively closed set generated by the image of S \ M leads to another pullback diagram. The result is that S M = T S\M L D D\{0} = T L k, where k denotes the quotient field of D. The basic gluing result for pullbacks ([10], Theorem 1.4) yields that the canonical map Spec(T ) Spec(S M ) is a homeomorphism in the Zariski topology. Note that the unique maximal ideal of S M is M, for M MS M MT = M. Hence Spec(S M ) = Spec(T ) as sets. Therefore, S M inherits the going-down domain property from T ([1], Proposition B.2). (b) By the proof of (a), Spec(S M ) = Spec(V ) as sets. Since V is a valuation domain, it follows that S M is a pseudo-valuation domain. We next present our main result. In effect, it generalizes Theorem 2.1 from the context of field extensions to a ring-theoretic setting. Theorem 2.5. If distinct going-down domains R T satisfy Spec(R) = Spec(T ), say with common maximal ideal M, then the following conditions are equivalent: (1) Each ring contained between R and T is a going-down domain; (2) Each ring contained between R and T is a treed domain; (3) Each ring contained between R/M and T/M is a going-down domain; (4) Each ring contained between R/M and T/M is a treed domain; (5) td R/M (T/M) 1. Proof. (1) (3), (2) (4): Each ring A contained between R/M and T/M is of the form A = S/M for some ring S between R and T. The assertions now follow from the fact that the class of going-down (resp., treed) domains is stable under the formation of factor domains ([5], Remarks 2.11 and 2.12 (a), (b)). (1) (2), (3) (4): Each going-down domain is treed. (3) (5): Apply Theorem 2.1. (4) (5): If X and Y are elements of T/M that are algebraically independent over R/M, then A := (R/M)[X, Y ] is not treed, for (X, Y ) is a maximal ideal of A that contains the incomparable prime ideals XA and Y A. (5) (1): Let S be a ring between R and T. As in the proof of Lemma 2.4, we have that MS M = M Spec(S) and S = T L D, where D := S/M and L := T/M. To show that S = S + MS M is a going-down domain, it suffices, by ([7], Corollary 2.3), to show that both S/M and S M are going-down domains.

6 18 R. DOUGLAS CHATHAM AND DAVID E. DOBBS Observe, given (5), that Theorem 2.1 ensures that S/M is a going-down domain. Finally, S M is a going-down domain by Lemma 2.4(a), to complete the proof. Recall from ([8]) that a domain R is called a locally pseudo-valuation domain (LP V D) if R P is a P V D for each P Spec(R). The most familiar examples of LP V Ds are P V Ds and Prüfer domains. For the sake of completeness, we include a role for LP V Ds in Corollary 2.6 (b). Of course, Corollary 2.6 (a) is our titular result. Corollary 2.6. Let (R, M) be a P V D, with canonically associated valuation overring V. Put k := R/M and L := V/M. Then: (a) Each overring of R is a going-down domain if and only if td k (L) 1. (b) The following four conditions are equivalent: (1) Each overring of R is a P V D; (2) Each overring of R is an LP V D; (3) L is algebraic over k; (4) V is the integral closure of R. Proof. (a) Using the fact that V is a valuation domain, one can adapt the proof of ([2], Theorem 3.1) to show that each overring of R is comparable with V (under inclusion). Also, each overring of V is a valuation domain, hence a Prüfer domain, hence a going-down domain. Therefore, it suffices to show that each ring contained between R and V is a going-down domain if and only if td k (L) 1. This follows, in turn, by applying Theorem 2.5. (b) Since V is integrally closed, (4) is equivalent to the condition that V is the integral closure of R in V. Therefore, by applying ([10], Corollary 1.5 (5)) to the pullback description R = V L k, we infer that (3) (4). However, (1) (4) (cf. ([12], Proposition 2.7)); and (2) (4) ([8], Corollary 2.10), to complete the proof. We close with a pair of examples of interesting P V D-theoretic behavior. Example 2.7. (a) For each n, 0 n, there exist distinct domains R T such that A is an n-dimensional P V D for all rings A such that R A T. (b) There exists a P V D, R, such that each overring of R is a going-down domain but some overring of R is not a P V D. Proof. (a) Consider any proper algebraic field extension k L such that L = T/M for some n-dimensional valuation domain (T, M). Then R := T L k T have the asserted properties. Indeed, if A is a ring between R and T, then D := A/M is a field (since L is algebraic over k), whence A = T L D is a P V D

7 ON PVDS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS 19 with canonically associated valuation overring T. Then Spec(A) = Spec(T ) as sets and, in particular, dim(a) = dim(t ) = n. (b) By Corollary 2.6 (a), (b), it suffices to choose R to be a P V D whose canonical pullback description, R = V L k, is such that td k (L) = 1. Perhaps the simplest such ring is R = [[k, k(y ) )) = k + Xk(Y )[[X]]; for this P V D, an example of a non-p V D overring of R is given by A = [[k[y ], k(y ) )) = k[y ] + Xk(Y )[[X]]. References [1] D. F. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math., 32 (1980), [2] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Mich. Math. J., 20 (1973), [3] R. D. Chatham, Going-down pairs of commutative rings, Ph. D. dissertation, University of Tennessee, Knoxville, [4] D. E. Dobbs, On going-down for simple overrings, II, Comm. Algebra, 1 (1974), [5] D. E. Dobbs, Divided rings and going-down, Pac. J. Math., 67 (1976), [6] D. E. Dobbs, Coherence, ascent of going-down and pseudo-valuation domains, Houston J. Math., 4 (1978), [7] D. E. Dobbs, On Henselian pullbacks, in Factorization in Integral Domains, Lecture Notes in Pure Appl. Math., 189 (1997), Dekker, New York, [8] D. E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl., 134 (1983), [9] D. E. Dobbs and I. J. Papick, On going-down for simple overrings, III, Proc. Amer. Math. Soc., 54 (1976), [10] M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl., 123 (1980), [11] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pac. J. Math., 75 (1978), [12] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, II, Houston J. Math., 4 (1978), [13] I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, [14] J.-P. Serre, Algèbre Locale. Multiplicités, Lecture Notes in Math., 11, Springer-Verlag, Berlin, Received August 18, 2000 (R. Douglas Chatham) Department of Mathematics, Wake Forest University, Winston- Salem, NC address: rdchat@math.wfu.edu (David E. Dobbs) Department of Mathematics, University of Tennessee, Knoxville, TN address: dobbs@math.utk.edu