Maximal non-Jaffard subrings of a field

Publicacions Matem`atiques, Vol. 44 (2000), 157–175 MAXIMAL NON-JAFFARD SUBRINGS OF A FIELD Mabrouk Ben Nasr and Noˆ oman Jarboui Abstract A domain Ris called a maximal non-Jaffard subring of a field L if R⊂L,Ris not a Jaffard domain and each domain Tsuch that R⊂T⊆Lis Jaffard. We show that maximal non-Jaffard subrings Rof a field Lare the integrally closed pseudo-valuation domains satisfying dimvR= dim R+1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when Ris integrally closed. Moreover, these domains are characterized in terms of the altitude formula in case Ris not integrally closed. An example of a maximal non-universally catenarian subring of its quotient field which is not integrally closed is given (Example 4.2). Other results and applications are also given. 0. Introduction Throughout this paper, R→Sdenotes an extension of commutative integral domains, qf(R) the quotient field of an integral domain R,R the integral closure of Rin its quotient field, and tr.deg[S:R] the transcendence degree of qf(S) over qf(R). If tr.deg[S:R]=0,we sayasin[4] that Sis algebraic over R. We recall that a ring Rof finite (Krull) dimension nis a Jaffard ring if its valuative dimension (the limit of the sequence (dim R[X1,... ,X n]−n,n∈N)) dimvR,is also n.Pr¨ufer domains and Noetherian domains are Jaffard domains. The notion of Jaffard ring is not a local property and thus we say that Ris a locally Jaffard ring if RPis a Jaffard ring for each prime ideal P of R. We assume familiarity with these concepts as in [1], [6], [14]. 1991 Mathematics Subject Classification. Primary 13B02; Secondary 13C15, 13A17, 13A18, 13B25, 13E05. Key words. Jaffard domain, Krull dimension, valuation domain, strong S-domain, pullbacks. 158 M. Ben Nasr, N. Jarboui A. R. Wadsworth in [22] considered pairs of domains R⊂Ssuch that each domain Tbetween Rand S(that is R⊆T⊆S) is Noetherian. In [21], S. Visweswaran noticed that if S=k[y1,... ,y t] is an affine domain (where kis a field) having Krull dimension n>0 and if Iis a nonzero proper ideal of Sand Da subring of k, then the ring R=D+Imay be not Noetherian, but each ring Tsuch that R⊂T⊆Sis Noetherian. In view of this, he introduced the following definition: “Let Abe a subring of a Noetherian ring B. Then Ais said to be a maximal nonNoetherian subring of Bif Ais non-Noetherian and any subring of B that properly contains Ais Noetherian”. S. Visweswaran characterized when D+Iis a maximal non-Noetherian subring of k[y1,... ,y t]([21, Proposition 2.1]). On the other hand, A. Ayache and P.-J. Cahen in [4] studied the domains Rsuch that each domain contained between Rand its quotient field qf(R) is Jaffard; that is, the domains whose integral closure RisaPr¨ufer domain ([4,Th´eor`eme 2.6]). They are said to be domains satisfying absolutely the altitude inequality formula. Our purpose here is to complete this circle of ideas by dealing with maximal non-Jaffard subrings of a field L; that is, the domains Rwhere Ris a nonJaffard domain and each ring T,R⊂T⊆Lis Jaffard. First, we show that if Ris a maximal non-Jaffard subring of a field L, then L= qf(R). Hence, we may restrict ourselves to the case where L= qf(R). Our concern in Section 1 is primarily with maximal non-Jaffard subrings. Our initial line of inquiry was suggested by pointing out a relationship between this kind of domains and pseudo-valuation domains (for short PVDs), which are closely related to valuation domains and have been deeply studied by several authors, notably J. R. Hedstrom, E. G. Houston [17], D. E. Dobbs and M. Fontana [13]. The main result of Section 1 is Theorem 1.4, which states that Ris a maximal nonJaffard subring of qf(R) if and only if Ris an integrally closed PVD and dimvR= dim R+ 1. Among the several interesting consequences ot this theorem, we only point out that if Ris a maximal non-Jaffard subring of qf(R), then Ris a residually integrally closed domain (in the sense that R/P is integrally closed for each prime ideal Pof R). The last part of this section is devoted to a deeper study of these rings, for instance, Theorem 1.7 states that Ris a maximal non-Jaffard subring of qf(R) if and only if Ris a local non-Jaffard domain with nonzero maximal ideal Mand for each ring Tsuch that R⊂T⊆qf(R) and for each prime ideal Qof T,ifQ=M, then R/(Q∩R)⊂T/Q is an algebraic extension, and if Q=M, then tr.deg[T/M :R/M] = 1. If Pis a property which may be possessed by rings (extensions) such as locally (totally) Jaffard, (stably) strong S, universally catenarian, we say that Maximal non-Jaffard subrings of a field 159 Ris a maximal non-Psubring of a field L,ifRis a non-Pdomain and each ring Tsuch as R⊂T⊆Lis P. We prove a result analogous to [6, Th´eor`eme 5.1]; we show in Theorem 1.8 that the following statements are equivalent: (i) Ris a maximal non-Jaffard subring of qf(R); (ii) Ris a maximal non-locally Jaffard subring of qf(R); (iii) Ris a maximal non-totally Jaffard subring of qf(R); (iv) Ris a maximal non-strong Ssubring of qf(R); (v) Ris a maximal non-stably strong Ssubring of qf(R). Recall from [4] that a domain Rsatisfies absolutely the altitude formula; if each overring of Rsatisfies the altitude formula, equivalently, each overring of Ris universally catenarian. Of course these domains satisfy absolutely the altitude inequality formula. However, the converse is not true [4, Exemple 5.1]. Section 2 is concluded with the study of maximal non-universally catenarian subrings of their quotient field. We begin by showing that this class of rings is larger than the class of domains which are maximal non-Jaffard subrings of their quotient fields; and if Ris assumed to be integrally closed, then these two classes of domains coincide. As Example 4.2 reveals, Theorem 2.3 is the best possible. This example depends ultimately on the pullback techniques in [15]. It points out the importance of the “integrally closed” hypothesis in Proposition 2.2 and suggests the need of characterizing the maximal non-universally catenarian subrings R, when Ris not integrally closed. We prove the following (Theorem 2.3): Let Rbe a non integrally closed domain. Then Ris a maximal nonuniversally catenarian subring of qf(R)if and only if Ris a Pr¨ufer domain, Rdoes not satisfy the altitude formula, and the extension T⊆ Tsatisfies the altitude formula for each proper overring Tof R. Section 3 finds necessary and sufficient conditions for certain pullbacks to be maximal non-Jaffard (resp., non-universally catenarian) subrings of their quotient fields. Section 4 is devoted to the investigation of several examples illuminating the earlier sections’ results, which in some cases are shown to be the best-possible. It cannot be expected that the typical reader is conversant with all this article’s references. Therefore in order to shorten this introduction, we have chosen to recall relevant definitions and facts as needed throughout the article. Any unexplained material is standard as in [16], [19]. 160 M. Ben Nasr, N. Jarboui 1. Maximal non-Jaffard subrings of a field Let Rbe a domain contained in a field L. We say that Ris a maximal non-Jaffard subring of Lif Ris not Jaffard and each ring Tsuch that R⊂T⊆L, is Jaffard. First of all, we establish the following. Proposition 1.1. Let Rbe a domain and La field containing R.IfR is a maximal non-Jaffard subring of L, then L= qf(R). Proof: First notice that Lis algebraic over R. Indeed, if not then there exists an element Xof Ltrancendental over R. Hence each overring of R[X] became Jaffard, that is R[X] is a Pr¨ufer domain, which is impossible [4,Th´eor`eme 2.6]. Now our task is to prove that qf(R)=L. Assume that qf(R)⊂Land let α∈L\qf(R). Then αis algebraic over R. Thus there exists an element r∈Rsuch that rα is integral over R.ThusR⊂R[rα] is an integral extension. But R[rα] is a Jaffard domain. Hence, according to [1, Proposition 1.1], Ris a Jaffard domain, the desired contradiction to complete the proof. As a direct consequence of Proposition 1.1, the study of maximal nonJaffard subrings of a field Lcan be reduced to the case where L= qf(R). Now, notice that if Ris a maximal non-Jaffard subring of its quotient field, then Ris integrally closed. Indeed, if R=R, then Ris a Jaffard domain, and hence so is R(since R⊂Ris an integral extension [1, Proposition 1.1]) which is impossible. In this section, we collect more information on this kind of domains and we characterize them in terms of pseudo-valuation domains. We begin by presenting some terminology. Let Bbe an integral domain, Ian ideal of B, and Da subring of B/I. Consider the pullback construction of commutative rings: R−−−−→D       B−−−−→B/I Following [9], we say that Ris the domain of the (B,I,D) construction and we set R:= (B,I,D). Next we consider the case where Iis assumed to be maximal. Denoting by Mthe ideal I,Tthe domain B,Kthe field T/M, and ϕ:T−→ Kthe natural epimorphism. Maximal non-Jaffard subrings of a field 161 We next recall a few wellknown properties about pullbacks to be used in our next theorems and examples. First, Mis a common ideal of both Rand T,M=(R:T)={x∈T|xT ⊂R}, and R/M ≃D.For each P∈Spec(R) with M⊆P, there is a (unique) Q∈Spec(D) such that ϕ−1(Q)=P; and ϕ−1(DQ)=RP.IfTis local, then each prime ideal of Rcompares with M, and thus dim R= dim D+ dim T. Also Ris local if and only if both Dand Tare local (cf. [15, Theorem 1.4, Proposition 2.1]). The last construction to be noted here concerns the notion of a pseudovaluation domain (for short, a PVD), which was introduced by Hedstrom and Houston [17] and has been studied subsequently in [2], [11], [12] and [18]. A domain Ris said to be a PVD in case each prime ideal Pof Ris strongly prime, in the sense that whenever x,y∈qf(R) satisfy xy ∈P, then either x∈Por y∈P, equivalently, in case Rhas a (uniquely determined) valuation overring Vsuch that Spec(R) = Spec(V) as sets, equivalently (by [2, Proposition 2.6]) in case Ris a pullback of the form V×Kk where Vis a valuation domain with residue field Kand kis a subfield of K. As the terminology suggests, any valuation domain is a PVD [17, Proposition 1.1]. Although the converse is false [17, Example 2.1], any PVD must, at least, be local [17, Corollary 1.3]. Before stating Theorem 1.4, we establish a lemma which serves both to motivate this theorem and to dispatch the difficult implication in its proof. First it is convenient to recall that given a ring extension R⊂S, (R, S) is said to be a Jaffard pair [7], if each ring Tbetween Rand Sis Jaffard. Lemma 1.2. Let k⊂Kbe an extension of fields. Then the following statements are equivalent. (i) (k,K)is a Jaffard pair; (ii) tr.deg[K:k]≤1. Proof: (ii)⇒(i) Assume that tr.deg[K:k]≤1 and let Tbe a ring between kand K. We have dimvT≤dimvk+tr.deg[T:k] (cf. [4, Lemme 1.1]). Thus dimvT≤1. If Tis not a field, then dim T= dimvT= 1, so Tis a Jaffard domain. (i)⇒(ii) If tr.deg[K:k]≥2, let X,Ybe two transcendental algebraically independent elements of Kover k. Then, the domain T= k+Yk(X)[Y](Y)is contained between kand K, and we have dim T=1 and dimvT=2[1, Proposition 2.5]. Thus Tis not a Jaffard domain, which contradicts assertion (i). Hence tr.deg[K:k]≤1. Now we establish the following usefull result. 162 M. Ben Nasr, N. Jarboui Lemma 1.3. Let Vbe a valuation domain with maximal ideal Mand residue field K.LetDbe a subring of Kand R:= (V,M,D). Then the overrings of Rare exactly the overrings of Vand the intermediate domains between Rand V. Proof: Let Sbe an overring of Rsuch that V⊆ S. Thus there exists v∈Vand v∈ S. Our task is to show that S⊆V. Let s∈S, assume that s∈ V. Since Vis a valuation domain, then 1 s∈M. Hence v s∈M. Therefore v=v ss∈S, the desired contradiction. As Lemma 1.3 is to be used repeatedly in the proof of some of the next theorems, we make use of it without any reference or comment. Theorem 1.4. Let Rbe a domain. Then the following statements are equivalent: (i) Ris a maximal non-Jaffard subring of qf(R); (ii) Ris an integrally closed PVD with maximal ideal Mand tr.deg[V/M :R/M]=1, where Vis the associated valuation domain of R; (iii) Ris an integrally closed PVD and dimvR= dim R+1. Proof: (i)⇒(ii) We have already observed that Ris integrally closed. Now we claim that there exists a valuation overring Vof Rand a nonzero prime ideal Qof Vsuch that R=RP+QVQ, where P=Q∩R. Indeed, assume that R⊂RP+QVQfor each valuation overring Vof Rand each nonzero prime ideal Qof V. Then RP+QVQis a Jaffard domain. Hence R⊂Vis a residually algebraic extension [1, Proposition 2.5 (b)]. Thus Ris a Pr¨ufer domain ([16, Theorem 19.15]), and therefore Ris a Jaffard domain. Now R=RP+QVQfor some valuation overring V of Rand nonzero prime ideal Qof V.ThusR:= (VQ,QV Q,R P/PRP). Hence Ris a PVD (cf. [2]). Denote QVQby M. Then Mis the unique maximal ideal of R. Notice that M=M∩R=QVQ∩R=Q∩R=P. Our task is to show that tr.deg[V/M :R/M] = 1. Let Dbe a domain such that R/M ⊂D⊂ V/M and consider the ring T:= (V,M,D). Then R⊂T⊂V,soTis a Jaffard domain. Hence Dis a Jaffard domain [1, Theorem 2.6]. Thus each intermediate domain between R/M and V/M is Jaffard. Hence by Lemma 1.2, we get tr.deg[V/M:R/M]≤1. The extension R/M ⊂V/M can not be algebraic since Ris not Jaffard [1, Proposition 2.5 (b)]. Thus tr.deg[V/M :R/M]=1. Maximal non-Jaffard subrings of a field 163 (ii)⇒(i) It is clear that Ris not a Jaffard domain [1, Proposition 2.5 (b)]. Now, let Tbe a domain such that R⊂T⊆qf(R). Then either Tis an overring of V, so it is Jaffard, or Tis an intermediate domain between Rand V,soT:= (V,M,D), where R/M ⊂D⊆V/M. According to Lemma 1.2, Dis a Jaffard domain, and since Ris integrally closed, then tr.deg[V/M :D] = 0. Thus Tis a Jaffard domain [1, Theorem 2.6]. Hence Ris a maximal non-Jaffard subring of qf(R). (ii)⇔(iii) We have dim R= dim Vand dimvR=dim V+tr.deg[V/M : R/M] (cf. [1, Proposition 2.5 (a)]). Thus dimvR= dim R+1 if and only if tr.deg[V/M :R/M]=1. We turn now to point out the connection with integrally closed domains. A special result is that if Ris a maximal non-Jaffard subring of qf(R), then R/P is integrally closed for each prime ideal Pof R.We begin by noticing that if Ris a maximal non-Jaffard subring of qf(R), then for each non-maximal prime ideal Pof R,RPis a valuation domain since V⊂RP. Nevertheless we have the following: Proposition 1.5. Let Rbe a maximal non-Jaffard subring of qf(R). Then for each non maximal prime ideal Pof R,R/P is a maximal non-Jaffard surbring of its quotient field. Proof: Notice that R/P is not a Jaffard domain. Indeed, consider the domain R+PRP.IfR/P is a Jaffard domain, then so is R+PRP [1, Theorem 2.6 (b)]. Thus R⊂R+PRP. But Ris a PVD, so R:= (V,M,k)([2]), where Vis the associated valuation domain and k= R/M. Since P⊂M, then there exists a prime ideal Qof Vsuch that RP=VQ.ThusPRP=QVQ=Q⊆M. Therefore R+PRP= R+Q⊆R+M=R. This contradicts the fact that R⊂R+PRP. Consequently R/P is not a Jaffard domain. Now, let Dbe a domain such that R/P ⊂D⊆qf(R/P) and consider the ring T:= (RP,PR P,D). Since R/P ⊂D⊆qf(R/P), then R⊂T⊆qf(R). Hence Tis a Jaffard domain, and thus so is D[1, Theorem 2.6]. Therefore R/P is a maximal non-Jaffard subring of its quotient field. Remark 1.6.By the previous proposition, we conclude that if Ris a maximal non-Jaffard subring of qf(R), then R/P is integrally closed for each prime ideal Pof R. The converse does not hold. More precisely, in Example 4.1 we construct for each n,m∈Nsuch that n+1≤m≤ 2n+1, n≥2 a residually integrally closed local domain Rsuch that dim R=n, dim R[X]=m, and Ris not a maximal non-Jaffard subring of its quotient field. 164 M. Ben Nasr, N. Jarboui The next result provides another characterization of a domain Rwhich is a maximal non-Jaffard subring of its quotient field. Theorem 1.7. Let Rbe domain. Then the following statements are equivalent: (i) Ris a maximal non-Jaffard subring of qf(R); (ii) Ris local non-Jaffard domain with nonzero maximal ideal M, and for each ring Tsuch that R⊂T⊆qf(R)and for each Q∈Spec(T):IfQ=M, then R/(Q∩R)⊂T/Q is algebraic, and if Q=M, then tr.deg[T/M :R/M]=1. Proof: (i)⇒(ii) By Theorem 1.4, Ris local since it is a PVD. Now let T be a domain such that R⊂T⊆qf(R). Then either Tis an overring of V, where Vis the associated valuation domain of Ror Tis an intermediate domain between Rand V. In the first case, let Q∈Spec(T) such that Q=M, then R/(Q∩R)⊂V/(Q∩V)⊂T/Q. But tr.deg[V/(Q∩V): R/(Q∩R)] = 0 since Q∩R=M. Hence tr.deg[T/Q :R/(Q∩R)] = tr.deg[T/Q :V/(Q∩V)] + tr.deg[V/(Q∩V):R/(Q∩R)] = 0. Now, if Q=M, then T=V. Hence tr.deg[T/M :R/M]=tr.deg[V/M : R/M] = 1 (by Theorem 1.4). In the second case, Tshares the ideal M with Rand V.IfQ∈Spec(T) and Q=M, then we discuss two cases: Case 1: If M⊆ Q, then TQ=RQ∩R,soR/(Q∩R)⊂T/Q is algebraic. Case 2: If M⊂Q, then R+QTQ=R,soR+QTQis a Jaffard domain. Thus R/(Q∩R)⊂T/Q is an algebraic extension ([1, Theorem 2.6]). Now, if Q=M, then tr.deg[T/M :R/M]=tr.deg[V/M :R/M]− tr.deg[V/M :T/M]=1. (ii)⇒(i) Let Tbe a domain such that R⊂T⊆qf(R), and let Sbe an overring of T.IfQ∈Spec(S), then either Q=M, so by hypothesis, we have tr.deg[S/Q :R/(Q∩R)] = 0. Thus tr.deg[S/Q :T/(Q∩T)] = 0. Or Q=M,sotr.deg[S/M :R/M] = 1. Hence tr.deg[S/M :T/M]= tr.deg[S/M :R/M]−tr.deg[T/M :R/M] = 0. Therefore T⊆S is a residually algebraic extension for each overring Sof T.ThusT satisfies absolutely the altitude inequality formula [4,Th´eor`eme 2.6]. In particular, Tis a Jaffard domain. Thus Ris a maximal non-Jaffard subring of qf(R). Recall that a ring Ris said to be a strong S-domain if for each pair of consecutive prime ideals P⊂Qof R, the extended primes P[X]⊂ Q[X] are consecutive. The most natural examples of strong S-domains are arbitrary Noetherian domains [19, Theorem 68]. Dispite the above Maximal non-Jaffard subrings of a field 165 material, the class of strong S-domains is not very stable, for instance with respect to polynomial extensions. Following [20], we say that Ris a stably strong S-domain if R[X1,... ,X n] is a strong S-domain for each nonnegative integer n. A domain Ris said to be totally Jaffard if R/P is a locally Jaffard domain for each prime ideal Pof R(cf. [10]). We establish the following result. Theorem 1.8. Let Rbe a domain. Then the following statements are equivalent: (i) Ris a maximal non-Jaffard subring of qf(R); (ii) Ris a maximal non-locally Jaffard subring of qf(R); (iii) Ris a maximal non-totally Jaffard subring of qf(R); (iv) Ris a maximal non-strong Ssubring of qf(R); (v) Ris a maximal non-stably strong Ssubring of qf(R). Proof: It follows readily from [6,Th´eor`eme 5.1]. 2. Maximal non-universally catenarian subrings of their quotient fields First recall that an extension R⊆Sof integral domains satisfies the altitude inequality formula (resp., the altitude formula), if for each prime ideal Qof S,ifwesetP=Q∩R, we have htQ +tr.deg[S/Q :R/P]≤ htP +tr.deg[S:R] (resp., htQ +tr.deg[S/Q :R/P]=htP +tr.deg[S: R]). A domain Ris said to satisfy the altitude inequality formula (resp., the altitude formula) if R⊆Ssatisfies the altitude inequality formula (resp., the altitude formula) for each finite type R-algebra Scontaining R. A ring Ris said to be catenarian in case, for each pair P⊂Qof prime ideals of R, all saturated chains of primes from Pto Qhave a common finite length. We shall say that Ris universally catenarian if the polynomial rings R[X1,... ,X n] are catenarian for each positive integer n[8]. Notice that if Ris universally catenarian, then it satisfies the altitude formula [8, Theorem 5.1]. In [4], A. Ayache and P.-J. Cahen studied the domains Rsuch that each domain Tbetween Rand qf(R) satisfies the altitude formula. These domains are said to satisfy absolutely the altitude formula. They established that Rsatisfies absolutely the altitude formula if and only if RisaPr¨ufer domain and R⊆Rsatisfies the altitude formula ([4, Th´eor`eme 3.3]). Naturally these domains satisfy absolutely the altitude inequality formula, but the converse does not hold (cf. [4, Exemple 5.1]). 172 M. Ben Nasr, N. Jarboui Mn)/(Ms+···+Mn)≃Ds−1, which is integrally closed. If we assume that m>n+1,Rcan not be a maximal non-Jaffard subring of qf(R). Since if not, then we get dimvR= dim R+1=n+ 1 (Theorem 1.4). But dim R[X]=m>n+ 1 = dimvR, which is impossible. The next example illustrates the fact that Proposition 2.2 does not hold if Ris not integrally closed. Example 4.2. This example provides a domain Rsuch that: (a) Ris not integrally closed. (b) Rsatisfies absolutely the altitude inequality formula. (c) Ris a maximal non-universally catenarian subring of its quotient field. Let V,Wbe two incomparable valuation domains with the same quotient field L, with maximal ideals M1and N1, respectively. Assume that dim V= 2, dim W= 1, and that V/M1≃W/N1≃K.Thus S=V∩Wis a semilocal Pr¨ufer domain with maximal ideals M=M1∩S and N=N1∩S. Set R:= (S, I, K), where I=M∩N. We have dim S= 2 and dim R= 1 (cf. [9, Corollaire 2]). The integral closure R of Ris equal to S[9, Proposition 2]. Hence Ris a Pr¨ufer domain. Thus Rsatisfies absolutely the altitude inequality formula [4,Th´eor`eme 2.6]. We have htSN+tr.deg[S/N :R/(N∩R)] = 1 <ht R(N∩R)+ tr.deg[S:R]=htRI= 2. Thus the finite type extension R⊂Sdoes not satisfy the altitude formula. Therefore Rdoes not satisfy the altitude formula. Now let Tbe a domain such that R⊂T⊆qf(R). Our task is to show that Tsatisfies the altitude formula. We claim that Tis a Pr¨ufer domain (since it is an overring of R). By [14, Proposition 2.6], it will suffices to show that the extension T⊆Tsatisfies the altitude formula. Notice that Tis an overring of S; then T∈{S, V, W, VQ 1,V Q 1∩W, qf(S)}([16]), where Q 1is the unique prime ideal of Ssuch that (0) ⊂Q 1⊂M. Let Qbe a nonzero prime ideal of T. Set Q1=Q∩S,Q=Q∩T, and P=Q∩R. Our task is to show that htQ=htQ. Two cases are then possible: Case 1: If Q1=N, then htQ1=htP. On the other hand, htQ= htQ1since the extension S⊆Tis residually algebraic and satisfies the altitude formula. Thus htQ=htP. The domain Ris locally Jaffard; hence the extension R⊂Tsatisfies the altitude inequality formula ([4,Th´eor`eme 1.5]), in particular, htQ ≤htP ; which yields htQ ≤htQ. But T⊂Tis an integral extension, so it satisfies INC. Thus htQ≤htQ. Therefore htQ=htQ. Maximal non-Jaffard subrings of a field 173 Case 2: If Q1=N, then Q1is a maximal ideal of S. Hence Qis a maximal ideal of T. We discuss the following cases: (i) If T=W, then Tis local with dim T= dim T=htQ= 1. Thus Qis a maximal ideal of T. Hence htQ=htQ. (ii) The case when either T=Vor T=VQ 1is impossible since Ndoes not lift in V. (iii) If T=VQ 1∩W, then Tis semilocal with maximal ideals Q 1T Q 1and NT N. Thus dim T=1=htQ, and since dim T= dim T= 1, then htQ =htQ= 1. (iv) If T=S, in this case we have the following inclusions R⊂T⊆ T=S.ThusR/I ⊂T/I ⊆S/I; that is K⊂T/I ⊆K×K. Then necessarily we get T/I =K×K. Indeed, K×Kis a Kvector space of dimension 2. Thus T/I is K-vector space of dimension at most 2, and since T/I =K, then T/I =K×K. Therefore T=S=Tand Q=Q. As stated earlier, if we leave out the assumption “Tis a valuation domain” in Theorem 3.4, the following example shows, among other facts that the implication (ii)⇒(i) fails. Example 4.3. Denote by Dthe domain Rconstructed in Example 4.2. Let L= qf(D) and X,Ybe two indeterminates over L. Set V1= L(Y)+XL(Y)[X](X)and V2=L+YL[Y](Y)+(X+1)L(Y)[X](X+1). Then V1and V2are two incomparable valuation domains with maximal ideals M1=XL(Y)[X](X)and M2=YL[Y](Y)+(X+1)L(Y)[X](X+1), respectively. We have dim V1= 1 and dim V2= 2. The domain T1= V1∩V2is a semilocal Pr¨ufer domain with maximal ideals M=M1∩T1 and N=M2∩T1. Let M=M∩Nand T:= (T1,M,L). Let R:= (T,M,D). Then we have the following properties: (α)Dis a Jaffard domain which is a maximal non-universally catenarian subring of its quotient field. (See Example 4.2). (β)Tis a Pr¨ufer domain since T=T. (γ)D⊂T/M is an algebraic extension. (δ)Tdoes not satisfy the altitude formula (because the finite type extension T⊂T1does not satisfy the altitude formula). Hence R is not a maximal non-universally catenarian subring of its quotient field. 174 M. Ben Nasr, N. Jarboui References [1] D. F. Anderson, A. Bouvier, D. E. Dobbs, M. Fontana and S. Kabbaj, On Jaffard domains, Exposition. Math. 5(1988), 145–175. [2] D. F. Anderson and D. E. 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Houston, Pseudo-valuation domains, Pacific J. Math. 75(1) (1978), 137–147. [18] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, II, Houston J. Math. 4(2) (1978), 199–207. [19] I. Kaplansky,“Commutative rings”, Revised edition. The University of Chicago Press, Chicago, Ill.-London, 1974. [20] S. Malik and J. L. Mott, Strong S-domains, J. Pure Appl. Algebra 28(3) (1983), 249–264. [21] S. Visweswaran, Intermediate rings between D+Iand K[y1,... ,y t], Comm. Algebra 18(2) (1990), 309–345. [22] A. R. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc. 195 (1974), 201–211. Department of Mathematics Faculty of Sciences University of Sfax 3038 Sfax Tunisia E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 9 de febrer de 1999, darrera versi´o rebuda el 8 de juliol de 1999.