Research Article | Open Access
Transfer of the GPIT Property in Pullbacks
Let be a commutative ring, a prime ideal of , a subring of , and the pullback . Ascent and descent results are given for the transfer of the -PIT and GPIT (generalized principal ideal theorem) properties between and . As a consequence, it follows that if is a maximal ideal of both and , then satisfies -PIT (resp., GPIT) if and only if satisfies -PIT (resp., GPIT).
All rings considered in this paper are commutative, with identity. If is a ring, denotes the set of all prime ideals of , and if , then denotes the height of in . As in , if is a nonnegative integer, then we say that a ring satisfies the - property if whenever is minimal as a prime ideal of containing a given -generated ideal of . Notice that each ring satisfies 0- and that 1- is the (for “principal ideal theorem") property introduced in . Also as in , we say that a ring satisfies the GPIT (for “generalized principal ideal theorem") property in case satisfies - for all . 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 [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 and GPIT in pullbacks in case is a quasilocal domain and is a subring of (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 is not necessarily quasilocal or a domain. We thus enlarge the arena of applications in the same spirit in which the general construction studied by Brewer and Rutter  extended the classical construction (as in ) that had issued from a valuation (in particular, quasilocal) domain .
It is convenient to organize our work so that the assertions regarding transfer of GPIT are consequences of transfer results on -. 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 construction in Corollary 2.7.
Henceforth, we adopt the following standing hypotheses: is a ring, , is a subring of , and is the pullback . Before giving our main descent result for the - and GPIT properties, we begin with a lemma that builds on a result of Cahen [5, Proposition 5], that compares the heights of in and . 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 .
Lemma 2.1. Let , , , and be as in the standing hypotheses. Assume also that if satisfies , then either is comparable to (with respect to inclusion) or . Then, .
Proof. According to [5, Proposition 5], . Thus, without loss of generality, we may assume that . 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 such that and . By applying the isomorphism in [4, Corollary 1.5(3)], we see that there exists a (uniquely determined) such that and, moreover, that . In view of the hypothesis, there are three cases to consider.
If , then , a contradiction. On the other hand, if , then , also a contradiction. We handle the remaining case, in which , by an argument that is reminiscent of a proof of Cahen [5, Lemme 5]. In this case, , for some and . Then, and so , the desired contradiction.
Theorem 2.2. Let , , , and be as in the standing hypotheses. Assume that if satisfies , then either is comparable to or . Assume also that . Then,(a)if is a positive integer and satisfies -, then satisfies -;(b)if satisfies , then satisfies .
Proof. The assertion in (b) follows from that in (a), by universal quantification on . As for (a), consider an -generated ideal of and let be minimal among the prime ideals of that contain . Our task is to show that .
Suppose first that . Then, every prime of that is contained in is actually a prime ideal of , and so it follows easily that is minimal as a prime ideal of containing . Since satisfies -, . Therefore, by Lemma 2.1, .
In the remaining case, . Since , it follows that . An appeal to [4, Corollary 1.5(3)] yields a (uniquely determined) such that . Clearly, . Moreover, since is uniquely determined, it follows easily that is minimal among primes of that contain . As satisfies -, we have that . Furthermore, since , [4, Theorem 1.4(c)] ensures that . Accordingly
Recall (cf. ) that a ring is said to be treed in case no maximal ideal of contains prime ideals of that are incomparable.
Corollary 2.3. Let , , , and be as in the standing hypotheses. Assume also that and either or is treed. Then,(a)if is a positive integer and satisfies -, then satisfies -;(b)if satisfies , then satisfies .
Proof. The assumptions ensure that if , then either is comparable to or . An application of Theorem 2.2 completes the proof.
Lemma 2.4. Let , , , and be as in the standing hypotheses. Assume that if , then either is comparable to or . If and , then .
Proof. We adapt the proof of Lemma 2.1. Let such that . Since , we may assume that ; in particular, . Therefore, by [4, Corollary 1.4(3)], for some (uniquely determined) . In view of the hypothesis, there are three cases to consider.
The case is handled as in the proof of Lemma 2.1. On the other hand, if , then , and so . Finally, the ostensibly final case, , cannot actually arise, for otherwise, the proof of Lemma 2.1 would show that , a contradiction.
We next present our main ascent result for the - and properties.
Theorem 2.5. Let , , , and be as in the standing hypotheses. Assume also that . Then,(a)if is a positive integer and satisfies -, then satisfies -;(b)if satisfies , then satisfies .
Proof. As in the proof of Theorem 2.2, it suffices to establish (a). To that end, let be minimal as a prime ideal of containing a given -generated ideal . We consider two cases.
Suppose first that , that is, . Choose . We claim that is minimal as a prime ideal of containing the -generated ideal . If not, pick such that . By the minimality of , for some , and so since is prime. Then, , a contradiction, thus proving the above claim.
Since , it is clear that contains the elements . Moreover, it follows from the isomorphism in [4, Corollary 1.5(3)] and the minimality of that is minimal among the prime ideals of containing the -generated ideal . Furthermore, this isomorphism yields that and the assumption that satisfies - yields that . Thus, , as desired.
It remains only to consider the case . Since , Lemma 2.4 may be applied, with the upshot that . Suppose that and are such that . Then, by Lemma 2.4 and the minimality of , we have that and . Hence, is minimal as a prime ideal of containing . Since satisfies -, . Thus, , which completes the proof.
Corollary 2.6. Let , , , and be as in the standing hypotheses. Assume also that and . Then(a)for any positive integer , satisfies - if and only if satisfies -;(b) satisfies if and only satisfies .
Corollary 2.7. Let be a ring, where is a field and . Let be a subfield of , and put . Then,(a)for any positive integer , satisfies - if and only if satisfies -;(b) satisfies if and only satisfies .
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 is quasilocal. In those earlier results, the PIT (resp., GPIT) property for was shown to be equivalent to the condition that and the -PIT (resp., GPIT) property for . However, the condition that appears as a hypothesis, rather than a conclusion, in Theorem 2.2. Accordingly, we should underscore that, for each , the GPIT property for does not imply the -PIT property for under our standing hypothesis, even if .
To see this, begin by taking to be an -dimensional Noetherian unique factorization domain. Let denote the quotient field of . Set , with , and . Observe that the standing hypotheses are satisfied. Moreover, and, since is a Noetherian ring, satisfies GPIT (and, hence, -PIT). However, does not satisfy -PIT.
Indeed, pick a prime element of , so that by the classical principal ideal theorem. We have , essentially since . Moreover, , since . Then, the order-theoretic impact of the fundamental gluing result for pullbacks [4, Theorem 1.4] yields that , even though is an -generated ideal of , and so does not satisfy -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 and satisfies -PIT with , implies that . However, we do have the following positive result along these lines for the context. Let be a domain, where is a field and . Let be a subring of , and put . If satisfies PIT, then is a field.
For an indirect proof, suppose that we can pick a nonzero nonunit . Then, for all , we have , whence . Moreover, (the point being that since ). As is a nonunit of , we can choose minimal among the prime ideals of containing . Then, , contrary to satisfying PIT.
(c) We close by answering a question of the referee that asked for additional examples of a ring satisfying the hypothesis of Theorem 2.2. Specifically, we show that for each positive integer , there exists a nontreed non-quasilocal -dimensional domain and a nonmaximal nonzero prime ideal of such that, whenever satisfies , then either is comparable to (with respect to inclusion) or .
For the details of the construction, suppose first that . Then, a suitable can be found that possesses exactly prime ideals. Indeed, consider the -element poset , where the only nontrivial relations are given by if and only if ; ; and . It is known (cf. [9, Theorem 2.10]) that there exists a ring such that is order-isomorphic to , since is a finite poset. Then, taking to be the associated reduced ring of suffices, for the only prime ideal of that has height at least that of is , which is comparable to . Note that has exactly two maximal ideals, namely, and . The reader is invited to augment the above construction and thus produce an example with any desired finite number of maximal ideals.
Finally, in case , one can produce a suitable by arguing as above with the 6-element poset , where the only nontrivial relations are given by , and . 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 -Noetherian ring satisfies GPIT, while [10, Theorem 2.2] characterizes a -Noetherian ring as a -ring such that is the pullback of a certain type of diagram.
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Copyright © 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.