Transfer of the GPIT Property in Pullbacks

Abstract

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 𝑃∈𝑆𝑝𝑒𝑐(𝐴), then ℎ𝑡𝑡_𝐴(𝑃) denotes the height of 𝑃 in 𝐴. As in [1], if 𝑛 is a nonnegative integer, then we say that a ring 𝐴 satisfies the 𝑛-PIT property if ℎ𝑡𝑡_𝐴(𝑃)≤𝑛 whenever 𝑃∈𝑆𝑝𝑒𝑐(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 [2]. Also as in [1], we say that a ring 𝐴 satisfies the GPIT (for “generalized principal ideal theorem") property in case 𝐴 satisfies 𝑛-PIT for all 𝑛≥0. 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 [6] extended the classical 𝐷+𝑀 construction (as in [7]) 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 [7] or [3].

2. Results

Henceforth, we adopt the following standing hypotheses: 𝑇 is a ring, 𝐼∈𝑆𝑝𝑒𝑐(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 ℎ𝑡𝑡_𝑅(𝐼)=ℎ𝑡𝑡_𝑇(𝐼).

Lemma 2.1. Let 𝑇, 𝐼, 𝐷, and 𝑅 be as in the standing hypotheses. Assume also that if 𝑄∈𝑆𝑝𝑒𝑐(𝑇) satisfies ℎ𝑡𝑡_𝑇(𝑄)≥ℎ𝑡𝑡_𝑇(𝐼), then either 𝑄 is comparable to 𝐼 (with respect to inclusion) or 𝑄+𝐼=𝑇. Then, ℎ𝑡𝑡_𝑅(𝐼)=ℎ𝑡𝑡_𝑇(𝐼).

Proof. According to [5, Proposition  5], ℎ𝑡𝑡_𝑇(𝐼)≤ℎ𝑡𝑡_𝑅(𝐼). Thus, without loss of generality, we may assume that 𝑛∶=ℎ𝑡𝑡_𝑇(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 𝑞∈𝑆𝑝𝑒𝑐(𝑅) such that 𝑞⊂𝐼 and ℎ𝑡𝑡_𝑅(𝑞)≥𝑛. By applying the isomorphism in [4, Corollary  1.5(3)], we see that there exists a (uniquely determined) 𝑄∈𝑆𝑝𝑒𝑐(𝑇) such that 𝑄∩𝑅=𝑞 and, moreover, that ℎ𝑡𝑡_𝑇(𝑄)=ℎ𝑡𝑡_𝑅(𝑞)≥𝑛. In view of the hypothesis, there are three cases to consider.
If 𝑄⊇𝐼, then 𝑞=𝑄∩𝑅⊇𝐼∩𝑅=𝐼, a contradiction. On the other hand, if 𝑄⊂𝐼, then ℎ𝑡𝑡_𝑇(𝐼)≥ℎ𝑡𝑡_𝑇(𝑄)+1≥𝑛+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 𝛼∈𝑄 and 𝑖∈𝐼. Then, 𝛼=1−𝑖∈𝑄∩𝑅=𝑞⊂𝐼 (2.1) and so 1=𝛼+𝑖∈𝐼+𝐼⊆𝐼, 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_𝑇(𝑄)≥ℎ𝑡𝑡_𝑇(𝐼), then either 𝑄 is comparable to 𝐼 or 𝑄+𝐼=𝑇. Assume also that 𝐼∈Max(𝑅). Then,(a)if 𝑛 is a positive integer and 𝑇 satisfies 𝑛-PIT, then 𝑅 satisfies 𝑛-PIT;(b)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=∑^𝑛_𝑖=1𝑅𝑎_𝑖 of 𝑅 and let 𝑝 be minimal among the prime ideals of 𝑅 that contain 𝐽. Our task is to show that ℎ𝑡𝑡_𝑅(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=∑^𝑛_𝑖=1𝑇𝑎_𝑖. Since 𝑇 satisfies 𝑛-PIT, ℎ𝑡𝑡_𝑇(𝐼)≤𝑛. Therefore, by Lemma 2.1, ℎ𝑡𝑡_𝑅(𝑝)=ℎ𝑡𝑡_𝑅(𝐼)=ℎ𝑡𝑡_𝑇(𝐼)≤𝑛.
In the remaining case, 𝑝≠𝐼. Since 𝐼∈Max(𝑅), it follows that 𝐼⊈𝑝. An appeal to [4, Corollary  1.5(3)] yields a (uniquely determined) 𝑃∈𝑆𝑝𝑒𝑐(𝑇) such that 𝑃∩𝑅=𝑝. Clearly, 𝐼⊈𝑃. 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_𝑇(𝑃)≤𝑛. Furthermore, since 𝐼⊈𝑝, [4, Theorem  1.4(c)] ensures that 𝑅_𝑝=𝑇_𝑃. Accordingly ℎ𝑡𝑡_𝑅𝑝=ℎ𝑡𝑡_𝑅𝑝𝑝𝑅_𝑝=ℎ𝑡𝑡_𝑇𝑃𝑃𝑇_𝑃=ℎ𝑡𝑡_𝑇(𝑃)≤𝑛. (2.2)

Recall (cf. [8]) 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 𝐼∈Max(𝑅) and either 𝐼∈Max(𝑇) or 𝑇 is treed. Then,(a)if 𝑛 is a positive integer and 𝑇 satisfies 𝑛-PIT, then 𝑅 satisfies 𝑛-PIT;(b)if 𝑇 satisfies GPIT, then 𝑅 satisfies GPIT.

Proof. The assumptions ensure that if 𝑄∈𝑆𝑝𝑒𝑐(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 𝑄∈𝑆𝑝𝑒𝑐(𝑇), then either 𝑄 is comparable to 𝐼 or 𝑄+𝐼=𝑇. If 𝑃∈𝑆𝑝𝑒𝑐(𝑅) and 𝑃⊆𝐼, then 𝑃∈𝑆𝑝𝑒𝑐(𝑇).

Proof. We adapt the proof of Lemma 2.1. Let 𝑃∈𝑆𝑝𝑒𝑐(𝑅) such that 𝑃⊆𝐼. Since 𝐼∈𝑆𝑝𝑒𝑐(𝑇), we may assume that 𝑃⊂𝐼; in particular, 𝑃⊉𝐼. Therefore, by [4, Corollary  1.4(3)], 𝑃=𝑄∩𝑅 for some (uniquely determined) 𝑄∈𝑆𝑝𝑒𝑐(𝑇). In view of the hypothesis, there are three cases to consider.
The case 𝑄⊇𝐼 is handled as in the proof of Lemma 2.1. On the other hand, if 𝑄⊂𝐼, then 𝑄⊂𝑅, and so 𝑃=𝑄∩𝑅=𝑄∈𝑆𝑝𝑒𝑐(𝑇). Finally, the ostensibly final case, 𝑄+𝐼=𝑇, cannot actually arise, for otherwise, the proof of Lemma 2.1 would show that 1∈𝐼, 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 𝐼∈Max(𝑇). Then,(a)if 𝑛 is a positive integer and 𝑅 satisfies 𝑛-PIT, then 𝑇 satisfies 𝑛-PIT;(b)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=∑^𝑛_𝑖=1𝑇𝑎_𝑖. We consider two cases.
Suppose first that 𝑃≠𝐼, that is, 𝑃⊉𝐼. Choose 𝛼∈𝐼⧵𝑃. We claim that 𝑃 is minimal as a prime ideal of 𝑇 containing the 𝑛-generated ideal 𝐻∶=∑^𝑛_𝑖=1𝑇𝑎_𝑖𝛼. If not, pick 𝑄∈𝑆𝑝𝑒𝑐(𝑇) such that 𝐻⊆𝑄⊂𝑃. By the minimality of 𝑃, 𝑎_𝑖∉𝑄 for some 𝑖, and so 𝛼∈𝑄 since 𝑄 is prime. Then, 𝛼∈𝑃, a contradiction, thus proving the above claim.
Since 𝛼∈𝐼⊂𝑅, it is clear that 𝑝∶=𝑃∩𝑅 contains the elements 𝑎₁𝛼,…,𝑎_𝑛𝛼. 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 ∑^𝑛_𝑖=1𝑅𝑎_𝑖𝛼. Furthermore, this isomorphism yields that ℎ𝑡𝑡_𝑇(𝑃)=ℎ𝑡𝑡_𝑅(𝑝) and the assumption that 𝑅 satisfies 𝑛-PIT yields that ℎ𝑡𝑡_𝑅(𝑝)≤𝑛. Thus, ℎ𝑡𝑡_𝑇(𝑃)≤𝑛, as desired.
It remains only to consider the case 𝑃=𝐼. Since 𝐼∈Max(𝑇), Lemma 2.4 may be applied, with the upshot that ℎ𝑡𝑡_𝑇(𝐼)=ℎ𝑡𝑡_𝑅(𝐼). Suppose that 𝐺∶=∑^𝑛_𝑖=1𝑅𝑎_𝑖 and 𝑃₀∈𝑆𝑝𝑒𝑐(𝑅) are such that 𝐺⊆𝑃₀⊆𝑃=𝐼. Then, by Lemma 2.4 and the minimality of 𝑃, we have that 𝐽⊆𝑃₀∈𝑆𝑝𝑒𝑐(𝑇) and 𝑃₀=𝑃. Hence, 𝑃 is minimal as a prime ideal of 𝑅 containing 𝐺. Since 𝑅 satisfies 𝑛-PIT, ℎ𝑡𝑡_𝑅(𝑃)≤𝑛. Thus, ℎ𝑡𝑡_𝑇(𝑃)=ℎ𝑡𝑡_𝑇(𝐼)=ℎ𝑡𝑡_𝑅(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 𝐼∈Max(𝑅) and 𝐼∈Max(𝑇). Then(a)for any positive integer 𝑛, 𝑅 satisfies 𝑛-PIT if and only if 𝑇 satisfies 𝑛-PIT;(b)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 [6] in which 𝑇 need not be quasilocal.

Corollary 2.7. Let 𝐵=𝐾+𝑀 be a ring, where 𝐾 is a field and 𝑀∈Max(𝐵). Let 𝑘 be a subfield of 𝐾, and put 𝐴∶=𝑘+𝑀. Then,(a)for any positive integer 𝑛, 𝐴 satisfies 𝑛-PIT if and only if 𝐵 satisfies 𝑛-PIT;(b)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 𝐼∈Max(𝑅) and the 𝑛-PIT (resp., GPIT) property for 𝑇. However, the condition that 𝐼∈Max(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 𝐼∈Max(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, 𝐼∈Max(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 ℎ𝑡𝑡_𝐷(𝑝𝐷)=1 by the classical principal ideal theorem. We have 𝐼⊆𝑝𝑅, essentially since 𝑋=𝑝(𝑝⁻¹𝑋). Moreover, 𝑝𝑅∈𝑆𝑝𝑒𝑐(𝑅), since 𝑅/𝑝𝑅≅𝐷/𝑝𝐷. Then, the order-theoretic impact of the fundamental gluing result for pullbacks [4, Theorem  1.4] yields that ℎ𝑡𝑡_𝑅(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 𝐼∈Max(𝑇) and 𝑅 satisfies 𝑛-PIT with 𝑛>0, implies that 𝐼∈Max(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 0≠𝑀∈Max(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 𝑑∈𝐷. Then, for all 𝛼∈𝑀, we have 𝛼=𝑑(𝑑⁻¹𝛼)∈𝑑(𝐾𝑀)⊆𝑑𝑀⊆𝑑𝐴, whence 𝑀⊆𝑑𝐴. Moreover, 𝑀≠𝑑𝐴 (the point being that 𝑑∉𝑀 since 𝑀∩𝐷⊆𝑀∩𝐾=0). As 𝑑 is a nonunit of 𝐴, we can choose 𝑃 minimal among the prime ideals of 𝐴 containing 𝑑𝐴. Then, ℎ𝑡𝑡_𝐴(𝑃)≥ℎ𝑡𝑡_𝐴(𝑀)+1≥2, 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 𝑛≥2, there exists a nontreed non-quasilocal 𝑛-dimensional domain 𝑇 and a nonmaximal nonzero prime ideal 𝐼 of 𝑇 such that, whenever 𝑄∈𝑆𝑝𝑒𝑐(𝑇) satisfies ℎ𝑡𝑡_𝑇(𝑄)≥ℎ𝑡𝑡_𝑇(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 𝑛≥3. 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=𝑃_𝑛−1,𝑁=𝑃_𝑛,𝑀₁,𝑀=𝑀₂}, where the only nontrivial relations are given by 𝑃_𝑖<𝑃_𝑗 if and only if 0≤𝑖<𝑗≤𝑛; 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 𝑄≠𝐼 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 [10] 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.