DEMONSTRATIO MATHEMATICA Vol. XXXVI No 3 2003
Othman Echi, Noomen Jarboui
ON RESIDUALLY INTEGRALLY CLOSED DOMAINS
Abstract. A domain R is called residually integrally closed if R/p is an integrally closed domain for each prime ideal p of R. We show that residually integrally closed domains satisfy some chain conditions on prime ideals. We give characterization of such domains in case they contain a field of characteristic 0. Section 3 deals with domains R such that R/p is a unique factorization domain for each prime ideal p of R, these domains are showed to be PID. We also prove that domains R such that R/p is a regular domain are exactly Dedekind domains
1. Introduction All rings in this paper are assumed to be integral, commutative with identity, and the terminology is, in general, the same as that in [7], [8]. The quotient field of a ring R is denoted by qf(R), its Krull dimension by dimR. R' will always be used to denote the integral closure of a ring R in its quotient field. We denote by i?[n] the polynomial ring in n indeterminates over R. If p is a prime ideal of R, we denote by htp its height. Notice that if R is an integrally closed domain (even a UFD) and p is a prime ideal of R, then R/p may be not integrally closed. For instance, let R = Z[X],p = (X2 + 4)Z[X] and S := {a + 2bi \ a,b G Z and i2 = -1}. The domain R/p is isomorphic to 5; but it is not integrally closed since i € qf(S), i is integral over Z but i g S. Our concern in Section 2 is primarily with domains R such that R/p is integrally closed for each prime ideal p and we call them residually in- tegrally closed domains (for short RICD). In Proposition 2.1 we show that each integrally closed pseudo-valuation domain is a RICD. In Proposition 2.3 we show essentially that if R contains a field of characteristic 0, then
1991 Mathematics Subject Classification: Primary 13B02, 13522 13//05; Secondary 13C15; 13A18; 13B25; 13£05. Key words and phrases: integrally closed domain, unique factorization domain, regular domain. 544 O. Echi, N. Jarboui
R is a RICD if and only if for each prime ideal p of R and for each up- per P to p containing a monic polynomial, there exists a monic polynomial f(X) € such that P = Rad((p, f(X))R[X]). Proposition 2.6 gives another characterization of such domains which are supposed to be UFD. Recall that a domain R is said to be a unique factorization domain (or UFD), if each nonzero element of R is, in an essentially unique way (i.e., up to units), a product of irreducible ones. It is well known that a UFD is integrally closed. In spite of the simplicity of these notions, many problems concerning them have remained open for many years. For example the fact that every regular local ring is a UFD has been conjectured in the early 40's, many partial results in this direction have been proved, but the gen- eral case has been settled only in 1959 by M. Ausländer and D. Buchsbaum [1]. Section 3 is concluded with the study of residually unique factorization domains (for short RUFD) (resp., residually regular domains or RRD), that is, the domains R such that R/p is a UFD (resp., a regular domain) for each prime ideal p of R. Our initial line of inquiry was suggested by pointing out a relationship between this kind of domains and regular domains. More precisely, we prove that RUFD are precisely PID and RRD are Dedekind domains.
2. Residually integrally closed domains Let R be an integral domain. As noticed in the introduction, if R is an integrally closed domain and p is a prime ideal of R, then R/p may be not integrally closed. Our purpose in this section is to study the domains R such that R/p is integrally closed for each prime ideal p of R and we call them residually integrally closed domains (for short RICD). Notice that a one-dimensional integrally closed domain is a RICD. Recall from [6] that a domain R is said to be a pseudo-valuation domain in case each prime ideal p of R is strongly prime, in the sense that whenever x, y € qf(R) satisfy xy 6 p, then either x 6 p or y € p, equivalently, in case R has a (uniquely determined) valuation overring V such that Spec(R) = Spec(V) as sets equivalently (by [2, Proposition 2.6]) in case R is a pullback of the form Vx^fc, where V is a valuation domain with residue field K and k is a subfield of K. As the terminology suggests, any valuation domain is a pseudo-valuation domain [5, Proposition 1.1]. Although the converse is false [5, Example 2.1], any pseudo-valuation domain must, at least, be local [5, Corollary 1.3]. One may check easily that each valuation domain is a RICD, the following result is given for sake of examples.
PROPOSITION 2.1. Let R be a pseudo-valuation domain with associated val- uation domain V, then the following statements are equivalent: On residually integrally closed domains 545
(i) R is a RICD; (ii) R is integrally closed; (iii) k is algebraically closed in K. Proof. (i)=>-(ii). This is trivial. (ii)^(iii). See [3, Proposition 3], (ii)^(i). Denote by M the maximal ideal of R, k the residue field R/M. Let p be a nonzero prime ideal of R. If p = M, then R/p = R/M = k is obviously integrally closed. If p C M, then Rp = Vp [3, Proposition 0]. Hence qf(R/p) = qf(V/p) ~ Rp/pRp. Let x 6 Rp/pRp be an integral element over R/p. Then x is integral over V/p. Hence x € V/p since V/p is n integrally closed. Thus there exist ao, ai,..., a„_i 6 R such that x = an_i n 1 n n l x ~ + ... + ap; + ao. Hence x = an_ix ~ + ... + a\x + ao + r, where r € p. Therefore x is integral over R and as R is integrally closed we get x 6 R. So x e R/p which proves that R/p is integrally closed. • In what follows, we try to collect more information on residually inte- grally closed domains. We recall that if p is a prime ideal of R and if P is a prime ideal in the polynomial ring i?[X], then we call P an upper to p in R[X] if P n R = p, but P ± p[X\. In the following, we show that each RICD satisfies some chain conditions on prime ideals.
PROPOSITION 2.2. Let R be a RICD, then the following hold: (i) for each chain of primes of R : (0) C p C q and for each upper P to p such that P there exists an upper Q to (0) such that P is the unique upper to p containing Q and moreover Q q[X]; (ii) for each prime ideal p of R and for each upper P to p containing a monic polynomial, there exists a monic polynomial f(X) € R[X] such that P = Rad((p,f(X))R[X]). Proof, (i) Follows immediately from [12, (3.1)]. (ii) Let P be an upper to p containing a monic polynomial, then P/p[X] is an upper to (0) in (R/p)[X] containing a monic polynomial. Thus by [10, (2.4)], P/p[X] is generated by a monic polynomial f(X). Hence P = p[X] + f(X)R[X]. Thus P = Rad\(p, f(X))R[X]). » Let p be a nonzero prime ideal of R and Q an upper to (0). We denote by [Q, —> [p the set of all uppers P to p containing Q.
PROPOSITION 2.3. Let R be a domain containing a field of characteristic 0. Then the following statements are equivalent: (i) Ris a RICD; 546 O. Echi, N. Jarboui
(ii) for each prime ideal p of R and for each upper P to p containing a monic polynomial, there exists a monic polynomial f(X) E Z?[X] such that P = R*d((p,f(X))R[X}); (iii) for each prime ideal p of R and for each upper P to p containing a monic polynomial, there exists an upper Q to (0) containing a monic polynomial with [Q, —• [p= {P} and R is integrally closed. The proof of this Proposition breaks up into two lemmas. LEMMA 2.4 ([12, Lemma 3.3]). Let R be a local domain with maximal ideal M containing a field of characteristic 0. Let a E qf(R) such that a $ R and -1 a 0 R. Set fa : i2[X] —* R[oc] the epimorphism of R-algebras sending a to X. If f(X) is a polynomial contained in Kenpa but not contained in any other upper to (0) in except Kenpa, then f(X) E M[X], LEMMA 2.5. Let R be an integrally closed domain. Then the following state- ments are equivalent: (i) for each prime ideal p of R and for each upper P to p containing a monic polynomial, there exists a monic polynomial f(X) E such that P = Rad((p,f(X))R[X]); (ii) for each prime ideal p of R and for each upper P to p containing a monic polynomial, there exists an upper Q to (0) containing a monic polynomial with [Q, —• [p= {P}. Proof. (i)=>(ii). Let P be an upper to p containing a monic polynomial, there exists a monic polynomial f(X) € such that P = Rad((pJ(X))R[X]). Let Q be a prime ideal of with Q C P and with Q minimal over f(X)R[X], By [9, Lemma 3], Q is an upper to (0) containing a monic polyno- mial. We already have [Q,-> [p. Let P' € [Q, [p, then f(X) E Q C P' and p C P' so P = Rad((p, /(X))i?[X]) C P'. As P and P' are both uppers to p, P' = P. Thus [Q, [P = {P} so that (ii) holds. (ii)=>(i). Let P be an upper to p containing a monic polynomial. By (ii), there exists an upper Q to (0) containing a monic polynomial such that [Q, —> [p= {P}. Since R is integrally closed, there is a monic polyno- mial f(X) E such that Q = Rad(f(X)R[X}). We claim that P = Rad((p,f (X))R[X]). Let P' be a prime ideal minimal over (p, f(X))R[X]. By [9, Lemma 3] (used modulo p ), P' is an upper to p,. Since Q = Rad(f(X)R[X}) C P', we have P' e [Q,^ [p= {P}. Thus P' = P which proves the claim. • Proof of Proposition 2.3. (i)=»(ii). This follows from Proposition 2.2. (ii)^(iii). This follows readily from Lemma 2.5 On residually integrally closed domains 547
(ii)=r>(i). Assume that R/p is not integrally closed for some p 6 Spec(R). Let {R/p)' be the integral closure of R/p. Pick a € (R/p)' \ (R/p)• Let Q be the kernel of (the epimorphism of (i?/p)-algebras sending a to X)
PROPOSITION 2.6. Let R be a local UFD with maximal ideal M. Assume that R contains a field of characteristic 0. Then the following statements are equivalent: (i) Ris a RICD. (ii) for each chain of primes ofR : (0) C p C M and for each upper P top such that P M[X], there exists an upper Q to (0) such that [Q, —> [p= {P} and moreover Q M[X].
Proof. (i)=>(ii). See Proposition 2.2 (ii)=i>(i). This is the contrapositive of [12, (3.4)]. •
EXAMPLE 2.7. In the following, we construct for each integers n, m € N such that n+1
For this, write m = n + 1 + t. Observe that 0 < t < n. Let K = k(X i,X2, ...)> where k is a field and X\, X2, ...is an infinite number of indeterminates over k. Let Yi, ... Yt, Zi, ..., Zt, Zt+1, ..., Zn be indetermi- nates over K. Let DQ = K. For 1 < j < t, let Dj — Dj-I + Mj, where Mj = ZjK{Y\, Zi,..., Vj-i, Zj-i, Yj)[Zj](z) is the maximal ideal of the valuation domain Vi = K(Yi, Zh ..., ZjU, Yj)[Zj\z.y For t + 1 < j < n, let Dj = Dj-i+Mj, where Mj = ZjK(Yu Zh..., Yt'zt, Zt+1,..., Zj-^Zj]^ 548 O. Echi, N. Jarboui is the maximal ideal of the valuation domain
Vj = K(Yi, Zu..., Yu Zt, Zt+1,..., Zj-i)[Zj](Zj)- Notice that for any j such that 1 < j < n, the set of the nonzero prime ideals of Dj is {Ms+Ms+i + .. .+Mj \ s = 1,..., j} which is linearly ordered by inclusion. Observe also that Dj = K + Mi + ... + Mj. Let R = Dn. R is a local integral domain with maximal ideal M = M\+.. .+Mn. We have dimR = n and dimR[X] = m (cf. [3]). The domain R satisfies R/p is integrally closed for each p G Spec(R). Indeed, R is integrally closed [3, Proposition 2]. Let p be a nonzero prime ideal of R. There exists 1 < s < n such that p = Ms + Ms+1 + ... + M„. We have R/p = (Da-\ + Ma + ... + Mn)/{MS + ... + M„) ~ Ds-1, which is integrally closed [3],
3. RUFD and RRD Let (R, M) be an r-dimensional Noetherian local ring. If we set R/M = k, the smallest number of elements needed to generate M itself is equal to rankk(M/M2); (here rankk is the rank of a free module over k, that is the dimension of M/M2 as vector space). This number is called the embedding dimension of R, and is written embdimR. In general dimR < embdimR, and equality holds when M can be generated by r elements, in this case R is said to be a regular local ring, and a system of parameters generating M is called a regular system of parameters. A Noetherian domain R is said to be regular if Rp is a regular local domain for each prime ideal p of R. Recall that a unique factorization domain (or UFD) is an integral domain in which every element ^ 0 is, in an essentially unique way (i.e., up to units), a product of irreducible ones. It is well known that a regular local domain is a UFD [1] but the converse does not hold. We notice that if R is a UFD (resp., regular) and p is a prime ideal of Z?, then R/p may be not UFD (resp., regular). Thus we deal here with the study of domains R such that R/p is UFD (resp., regular) for each prime ideal p of R and we call them residually unique factorization domains (for short RUFD) (resp., residually regular domains or RRD). We establish the following two theorems characterizing RUFD and RRD. We are very grateful to Professor Luchezar Avramov for his help and com- ments concerning these results.
THEOREM 3.1. Let R be an integral locally finite dimensional domain. Then the following statements are equivalent: (i) Ris a RRD; (ii) R is a Dedekind domain. Proof. Clearly a Dedekind domain is a RRD. On residually integrally closed domains 549
(i)=*-(ii). Under the assumption (i), R is a Noetherian domain. Let us prove that R is a Dedekind domain. Indeed, then R = R/(0) is itself regular, so we need to prove that every prime ideal has height 1. If not, then R has a prime p of height 2. The local ring T = Rp is then regular of dimension 2. Every height 1 ideal q of T is principal, say q = (/). Choose one with / contained in the square of the maximal ideal of T (this is always possible). The local ring T/(f) then is not regular. It is also a localization of some R/p, so we have a contradiction. •
THEOREM 3.2. Let R be an integral locally finite dimensional domain. Then the following statements are equivalent: (i) R is a RUFD; (ii) for each consecutive primes of R : p C q, the ideal q/p is principal in R/p; (iii) R is a PID. Proof. (i)=i>(ii). Let p C q be two consecutive prime ideals of R, then q/p is a height 1 prime ideal of the UFD R/p. Hence q/p is principal in R/p. (ii)=i>(i). Obviously, condition (ii) is stable by passage to the quotient, hence it suffices to prove that R is a UFD. Since R is locally finite dimen- sional, every nonzero prime ideal in R contains a principal prime. Thus, following [7, Theorem 5], R is a UFD. (ii)=>-(iii) Under condition (i) or (ii), one may prove that R is RRD. For this end let p 6 Spec(R) and (0) C p\ C ... C pk = P be a saturated chain of prime ideals of R arising at p. Since htp\ = 1, then there exists an element ai € R such that p\ — a\R. Hence ai is a prime element. The ideal P2/P1 is principal in R/pi, thus there exists 02 € R such that P2/P1 = a^iR/pi)- Hence P2 = a^R + a\R. The same argument used permits one to construct ai, ü2, ..., afe 6 R such that pi = a\R + ... + aji? for each i E {1,..., k}. Hence p is generated by k elements of R. Hence R is regular. Therefore, following Theorem 3.1, R is a Dedekind domain. Finally, R is a Dedekind domain in which every maximal ideal is principal, so that R is a PID. (iii) (ii) A triviality. •
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DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES OF TUNIS UNIVERSITY TUNIS EL MANAR "CAMPUS UNIVERSITAIRE" 1060 TUNIS, TUNISIA E-mail: [email protected] FACULTY OF SCIENCES OF SFAX DEPARTMENT OF MATHEMATICS BP 802 3018 SFAX, TUNISIA E-mail: [email protected]
Received August 26, 2002.