IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation74387310.1155/2012/743873743873Research ArticleTransfer of the GPIT Property in PullbacksDobbsDavid E.1ShapiroJay2BadawiAyman1Department of MathematicsUniversity of TennesseeKnoxville, TN 37996-1320USAtennessee.edu2Department of MathematicsGeorge Mason UniversityFairfax, VA 22030-4444USAgmu.edu20121212012201206092011151120112012Copyright © 2012 David E. Dobbs and Jay Shapiro.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let T be a commutative ring, I a prime ideal of T, D a subring of T/I, and R the pullback T×T/ID. Ascent and descent results are given for the transfer of the n-PIT and GPIT (generalized principal ideal theorem) properties between T and R. As a consequence, it follows that if I is a maximal ideal of both T and R, then R satisfies n-PIT (resp., GPIT) if and only if T satisfies n-PIT (resp., GPIT).

1. Introduction

All rings considered in this paper are commutative, with identity. If A is a ring, Spec(A) denotes the set of all prime ideals of A, and if PSpec(A), then htA(P) denotes the height of P in A. As in , if n is a nonnegative integer, then we say that a ring A satisfies the n-PIT property if htA(P)n whenever PSpec(A) is minimal as a prime ideal of A containing a given n-generated ideal of A. Notice that each ring satisfies 0-PIT and that 1-PIT is the PIT (for “principal ideal theorem") property introduced in . Also as in , we say that a ring A satisfies the GPIT (for “generalized principal ideal theorem") property in case A satisfies n-PIT for all n0. A well-known fundamental result states that each Noetherian ring satisfies GPIT (cf. [3, Theorem  152]). Moreover, GPIT figures implicitly in a characterization of Noetherian rings [2, Theorem  2.6 and Remark  5.3(a)]. The diversity of rings satisfying PIT [2, Corollaries  3.5, 3.11, and 6.6] and GPIT [1, Corollaries 2.3 and 4.3] has been exhibited by studying the transfer of these properties for various ring-theoretic constructions (cf. [2, Propositions  3.1(a), 4.1 and 6.3, Theorem  4.5 and Section  5], [1, Proposition  2.1, Theorems  3.3 and 4.2]). In particular, in view of the characterizations of Noetherian rings arising as certain pullbacks ([4, Proposition  1.8], [5, Proposition  1, Corollaire  1]), earlier papers developed transfer results for PIT and GPIT in pullbacks D×T/IT in case (T,I) is a quasilocal domain and D is a subring of T/I (see [2, Corollary  3.2(c)], [1, Proposition  2.5(b)]). The main purpose of this note is to extend those results to the context in which T is not necessarily quasilocal or a domain. We thus enlarge the arena of applications in the same spirit in which the general D+M construction studied by Brewer and Rutter  extended the classical D+M construction (as in ) that had issued from a valuation (in particular, quasilocal) domain T.

It is convenient to organize our work so that the assertions regarding transfer of GPIT are consequences of transfer results on n-PIT. Our main descent (resp., ascent) result for these properties is Theorem 2.2 (resp., Theorem 2.5). Technical results dealing with new conditions on prime ideals in pullbacks are isolated in Lemmas 2.1 and 2.4. Our best “if and only if" transfer result appears in Corollary 2.6, with the special case for the D+M construction in Corollary 2.7.

In addition to the notation and terminology introduced above, we let Max(A) denote the set of all maximal ideals of a ring A. Any other material is standard, as in  or .

2. Results

Henceforth, we adopt the following standing hypotheses: T is a ring, ISpec(T), D is a subring of T/I, and R is the pullback T×T/ID. Before giving our main descent result for the n-PIT and GPIT properties, we begin with a lemma that builds on a result of Cahen [5, Proposition  5], that compares the heights of I in T and R. An additional hypothesis is used in Lemma 2.1, since an example of Cahen [5, Exemple  3], shows that the standing hypotheses are not sufficient to imply that htR(I)=  htT(I).

Lemma 2.1.

Let T, I, D, and R be as in the standing hypotheses. Assume also that if QSpec(T) satisfies htT(Q)htT(I), then either Q is comparable to I (with respect to inclusion) or Q+I=T. Then, htR(I)=htT(I).

Proof.

According to [5, Proposition  5], htT(I)htR(I). Thus, without loss of generality, we may assume that n:=  htT(I)<. The assertion can be proved by applying the fundamental gluing result on the spectra of pullbacks [4, Theorem  1.4]. We prefer the following somewhat more transparent argument.

Suppose that the assertion fails. Then, we can choose qSpec(R) such that qI and htR(q)n. By applying the isomorphism in [4, Corollary  1.5(3)], we see that there exists a (uniquely determined) QSpec(T) such that QR=q and, moreover, that htT(Q)=  htR(q)n. In view of the hypothesis, there are three cases to consider.

If QI, then q=QRIR=I, a contradiction. On the other hand, if QI, then htT(I)htT(Q)+1n+1, also a contradiction. We handle the remaining case, in which Q+I=T, by an argument that is reminiscent of a proof of Cahen [5, Lemme  5]. In this case, α+i=1, for some αQ and iI. Then, α=1-iQR=qI and so 1=α+iI+II, the desired contradiction.

Theorem 2.2.

Let T, I, D, and R be as in the standing hypotheses. Assume that if Q  Spec(T) satisfies htT(Q)htT(I), then either Q is comparable to I or Q+I=T. Assume also that IMax(R). Then,

if n is a positive integer and T satisfies n-PIT, then R satisfies n-PIT;

if T satisfies GPIT, then R satisfies GPIT.

Proof.

The assertion in (b) follows from that in (a), by universal quantification on n. As for (a), consider an n-generated ideal J=i=1nRai of R and let p be minimal among the prime ideals of R that contain J. Our task is to show that htR(p)n.

Suppose first that p=I. Then, every prime of T that is contained in I is actually a prime ideal of R, and so it follows easily that I is minimal as a prime ideal of T containing JT=i=1nTai. Since T satisfies n-PIT, htT(I)n. Therefore, by Lemma 2.1, htR(p)=  htR(I)=  htT(I)n.

In the remaining case, pI. Since IMax(R), it follows that Ip. An appeal to [4, Corollary  1.5(3)] yields a (uniquely determined) PSpec(T) such that PR=p. Clearly, IP. Moreover, since P is uniquely determined, it follows easily that P is minimal among primes of T that contain JT. As T satisfies n-PIT, we have that htT(P)n. Furthermore, since Ip, [4, Theorem  1.4(c)] ensures that Rp=TP. Accordingly htR(p)=htRp(pRp)=htTP(PTP)=htT(P)n.

Recall (cf. ) that a ring A is said to be treed in case no maximal ideal of A contains prime ideals of A that are incomparable.

Corollary 2.3.

Let T, I, D, and R be as in the standing hypotheses. Assume also that IMax(R) and either IMax(T) or T is treed. Then,

if n is a positive integer and T satisfies n-PIT, then R satisfies n-PIT;

if T satisfies GPIT, then R satisfies GPIT.

Proof.

The assumptions ensure that if QSpec(T), then either Q is comparable to I or Q+I=T. An application of Theorem 2.2 completes the proof.

We next isolate a lemma of some independent interest. Notice that the assumption in Lemma 2.4 arose naturally in the proof of Corollary 2.3.

Lemma 2.4.

Let T, I, D, and R be as in the standing hypotheses. Assume that if QSpec(T), then either Q is comparable to I or Q+I=T. If PSpec(R) and PI, then PSpec(T).

Proof.

We adapt the proof of Lemma 2.1. Let PSpec(R) such that PI. Since ISpec(T), we may assume that PI; in particular, PI. Therefore, by [4, Corollary  1.4(3)], P=QR for some (uniquely determined) QSpec(T). In view of the hypothesis, there are three cases to consider.

The case QI is handled as in the proof of Lemma 2.1. On the other hand, if QI, then QR, and so P=QR=QSpec(T). Finally, the ostensibly final case, Q+I=T, cannot actually arise, for otherwise, the proof of Lemma 2.1 would show that 1I, a contradiction.

We next present our main ascent result for the n-PIT and GPIT properties.

Theorem 2.5.

Let T, I, D, and R be as in the standing hypotheses. Assume also that IMax(T). Then,

if n is a positive integer and R satisfies n-PIT, then T satisfies n-PIT;

if R satisfies GPIT, then T satisfies GPIT.

Proof.

As in the proof of Theorem 2.2, it suffices to establish (a). To that end, let P be minimal as a prime ideal of T containing a given n-generated ideal J=i=1nTai. We consider two cases.

Suppose first that PI, that is, PI. Choose αIP. We claim that P is minimal as a prime ideal of T containing the n-generated ideal H:=i=1nTaiα. If not, pick QSpec(T) such that HQP. By the minimality of P, aiQ for some i, and so αQ since Q is prime. Then, αP, a contradiction, thus proving the above claim.

Since αIR, it is clear that p:=PR contains the elements a1α,,anα. Moreover, it follows from the isomorphism in [4, Corollary  1.5(3)] and the minimality of P that p is minimal among the prime ideals of R containing the n-generated ideal i=1nRaiα. Furthermore, this isomorphism yields that htT(P)=  htR(p) and the assumption that R satisfies n-PIT yields that htR(p)n. Thus, htT(P)n, as desired.

It remains only to consider the case P=I. Since IMax(T), Lemma 2.4 may be applied, with the upshot that htT(I)=  htR(I). Suppose that G:=i=1nRai and P0Spec(R) are such that GP0P=I. Then, by Lemma 2.4 and the minimality of P, we have that JP0Spec(T) and P0=P. Hence, P is minimal as a prime ideal of R containing G. Since R satisfies n-PIT, htR(P)n. Thus, htT(P)=  htT(I)=  htR(P)n, which completes the proof.

Combining Corollary 2.3 and Theorem 2.5, we obtain the following sufficient conditions for the GPIT property in R to be equivalent to the GPIT property in T.

Corollary 2.6.

Let T, I, D, and R be as in the standing hypotheses. Assume also that IMax(R) and IMax(T). Then

for any positive integer n, R satisfies n-PIT if and only if T satisfies n-PIT;

R satisfies GPIT if and only T satisfies GPIT.

We next state a special case of the previous corollary that generalizes [2, Corollary  3.2(c)] and [1, Proposition  2.5(b)] to the general D+M context of  in which T need not be quasilocal.

Corollary 2.7.

Let B=K+M be a ring, where K is a field and MMax(B). Let k be a subfield of K, and put A:=k+M. Then,

for any positive integer n, A satisfies n-PIT if and only if B satisfies n-PIT;

A satisfies GPIT if and only B satisfies GPIT.

Remark 2.8.

(a) It was stated in Section 1 that our results generalize transfer results for PIT and GPIT that had been given in [1, 2] for the case in which (T,I) is quasilocal. In those earlier results, the PIT (resp., GPIT) property for R was shown to be equivalent to the condition that IMax(R) and the n-PIT (resp., GPIT) property for T. However, the condition that IMax(R) appears as a hypothesis, rather than a conclusion, in Theorem 2.2. Accordingly, we should underscore that, for each n>0, the GPIT property for T does not imply the n-PIT property for R under our standing hypothesis, even if IMax(T).

To see this, begin by taking D to be an n-dimensional Noetherian unique factorization domain. Let L denote the quotient field of D. Set T:=L[X]=L+I, with I:=XT, and R:=T×T/ID=T×LD=D+XL[X]. Observe that the standing hypotheses are satisfied. Moreover, IMax(T) and, since T is a Noetherian ring, T satisfies GPIT (and, hence, n-PIT). However, R does not satisfy n-PIT.

Indeed, pick a prime element p of D, so that htD(pD)=1 by the classical principal ideal theorem. We have IpR, essentially since X=p(p-1X). Moreover, pRSpec(R), since R/pRD/pD. Then, the order-theoretic impact of the fundamental gluing result for pullbacks [4, Theorem  1.4] yields that htR(pR)=n+1, even though pR is an n-generated ideal of R, and so R does not satisfy n-PIT.

(b) Apropos of generalizing the pullback-theoretic results on PIT and GPIT in [1, 2], we do not know whether the standing hypotheses, coupled with the additional assumptions that IMax(T) and R satisfies n-PIT with n>0, implies that IMax(R). However, we do have the following positive result along these lines for the D+M context. Let B=K+M be a domain, where K is a field and 0MMax(B). Let D be a subring of K, and put A:=D+M. If A satisfies PIT, then D is a field.

For an indirect proof, suppose that we can pick a nonzero nonunit dD. Then, for all αM, we have α=d(d-1α)d(KM)dMdA, whence MdA. Moreover, MdA (the point being that dM since MDMK=0). As d is a nonunit of A, we can choose P minimal among the prime ideals of A containing dA. Then, htA(P)  htA(M)+12, contrary to A satisfying PIT.

(c) We close by answering a question of the referee that asked for additional examples of a ring T satisfying the hypothesis of Theorem 2.2. Specifically, we show that for each positive integer n2, there exists a nontreed non-quasilocal n-dimensional domain T and a nonmaximal nonzero prime ideal I of T such that, whenever QSpec(T) satisfies htT(Q)  htT(I), then either Q is comparable to I (with respect to inclusion) or Q+I=T.

For the details of the construction, suppose first that n3. Then, a suitable T can be found that possesses exactly n+3 prime ideals. Indeed, consider the (n+3)-element poset S={0=P0,P1,,I=Pn-1,N=Pn,M1,M=M2}, where the only nontrivial relations are given by Pi<Pj if and only if 0i<jn; P0<M1<M; and M1<N. It is known (cf. [9, Theorem  2.10]) that there exists a ring B such that Spec(B) is order-isomorphic to S, since S is a finite poset. Then, taking T to be the associated reduced ring of B suffices, for the only prime ideal QI of T that has height at least that of I is N, which is comparable to I. Note that T has exactly two maximal ideals, namely, N and M. The reader is invited to augment the above construction and thus produce an example T with any desired finite number 3 of maximal ideals.

Finally, in case n=2, one can produce a suitable T by arguing as above with the 6-element poset S={0,I,N,P,Q,M}, where the only nontrivial relations are given by 0<I<N,  0<P<M, and 0<Q<M. The simple verification in this case is left to the reader.

(d) We note that the paper  touches on some related matters. Indeed, [10, Theorem  2.7] shows that each ϕ-Noetherian ring satisfies GPIT, while [10, Theorem 2.2] characterizes a ϕ-Noetherian ring as a ϕ-ring R such that ϕ(R) is the pullback of a certain type of diagram.

AndersonD. F.DobbsD. E.Eakin,P. M.Jr.HeinzerW. J.On the generalized principal ideal theorem and Krull domainsPacific Journal of Mathematics199014622012151078378ZBL0746.13007BarucciV.AndersonD. F.DobbsD. E.Coherent Mori domains and the principal ideal theoremCommunications in Algebra19871561119115688294510.1080/00927878708823460ZBL0622.13007KaplanskyI.Commutative Rings1974Chicago, Ill, USAThe University of Chicago Pressix+1820345945FontanaM.Topologically defined classes of commutative ringsAnnali di Matematica Pura ed Applicata198012333135558193510.1007/BF01796550ZBL0443.13001CahenP.-J.Couples d'anneaux partageant un idéalArchiv der Mathematik198851650551497372510.1007/BF01261971ZBL0668.13005BrewerJ. W.RutterE. A.D+M constructions with general overringsThe Michigan Mathematical Journal19762313342040174410.1307/mmj/1029001619ZBL0318.13007GilmerR.Multiplicative Ideal Theory1972New York, NY, USAMarcel Dekkerx+609Pure and Applied Mathematics, No. 10427289DobbsD. E.On going down for simple overrings. IICommunications in Algebra19741439458036422510.1080/00927877408548715LewisW. J.The spectrum of a ring as a partially ordered setJournal of Algebra197325419434031481110.1016/0021-8693(73)90091-4ZBL0266.13010BadawiA.LucasT. G.Rings with prime nilradicalArithmetical Properties of Commutative Rings and Monoids2005241Boca Raton, Fla, USAChapman & Hall/CRC198212Lect. Notes Pure Appl. Math.2140694