In this article we show that any strongly separable extension of a commutative ring R can be embedded into another one having primitive element whenever every boolean localization of  R modulo its Jacobson radical is von Neumann regular and locally uniform

Bagio, D.

Paques, A.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Journal Algebra Discrete Math.Algebra and Discrete Mathematics RESEARCH ARTICLENumber 3. (2009). pp. 20 – 26c⃝ Journal “Algebra and Discrete Mathematics”A variant of the primitive element theorem forseparable extensions of a commutative ringDirceu Bagio and Antonio PaquesCommunicated by guest editorsAbstract. In this article we show that any strongly sep-arable extension of a commutative ring 푅 can be embedded intoanother one having primitive element whenever every boolean lo-calization of 푅 modulo its Jacobson radical is von Neumann regularand locally uniform.Dedicated to Professor Miguel Ferreroon occasion of his 70-th anniversaryIntroductionThroughout this paper by ring we mean a commutative ring with identityelement. By a connected ring we mean a ring whose unique idempotentsare 0 and 1. Furthermore, 퐽(푅) denotes the Jacobson radical of thering 푅.Given a ring extension 푆 ⊇ 푅 we say that 푆 has a primitive elementover 푅 if there exists 훼 ∈ 푆 such that 푆 = 푅[훼]. As it is well known,any ﬁnite separable ﬁeld extension has a primitive element. Althoughthis assertion is not true in general, several authors has been obtainedextensions of it for strongly separable extensions 푆 of a ring 푅. Forinstance, in the case that:— (푅,픪) is a local ring with∣∣푅픪∣∣ =∞ [6];This paper was partially supported by CAPES (Brazil)2000 Mathematics Subject Classification: 13B05, 12F10.Key words and phrases: primitive element, separable extension, boolean local-ization.Journal Algebra Discrete Math.D. Bagio, A. Paques 21— 푅 is a semilocal ring with∣∣푅픪∣∣ ≥ 푟푎푛푘푅푆, for every maximal ideal픪 of 푅 [10];— 푅 is a ring with many units and such that∣∣푅픪∣∣ ≥ 푟푎푛푘푅푆, for everymaximal ideal 픪 of 푅 [9].An alternative and also interesting question is to know under whatconditions the following variant of the primitive element theorem holds:(★) any strongly separable extension 푆 of a ring 푅 can be embedded intoanother one having primitive element.The statement (★) is true for connected strongly separable extensionsof 푅 in the following cases: 푅 is local [8, Theorem 1.1], 푅 is connected andsemilocal [1, Theorem 2.1.1] and 푅 is connected with 푅퐽(푅) von Neumannregular and locally uniform [2, Theorem 2.1].In this paper we show that the statement (★) is valid for any ring 푅,provided that 푅푥퐽(푅푥) is von Neumann regular and locally uniform, for allprime ideal 푥 in the boolean ring 퐵(푅) of all idempotents of 푅.1. PreliminariesIn this paper we will be employing freely the ideas and results of [12] onboolean spectrum and boolean localization of a ring (see also [7]). Webegin by recalling the terminology we will need.For any ring 푅 let 퐵(푅) denote the boolean ring of all idempotentsof 푅 and 푆푝푒푐(퐵(푅)) the boolean spectrum of 푅 consisting of all prime(equivalently maximal) ideals of 퐵(푅) (see [12, 2.1 and 2.2]). A base for atopology on 푆푝푒푐(퐵(푅)) is given by the family of basic open sets {푈푒∣푒 ∈퐵(푅)}, where 푈푒 = {푥 ∈ 푆푝푒푐(퐵(푅))∣1 − 푒 ∈ 푥}. This base deﬁnes acompact, totally disconnected, Hausdorﬀ topology on 푆푝푒푐(퐵(푅)).By localization of 푅 at 푥, for each 푥 ∈ 푆푝푒푐(퐵(푅)), we mean thequotient ring 푅푥 =푅퐼(푥) where 퐼(푥) denotes the ideal of 푅 generated bythe elements of 푥. By [12, 2.13] 푅푥 is a connected ring. For any 푅-module푀 , we set 푀푥 = 푀 ⊗푅 푅푥 =푀퐼(푥)푀 . For any element 푎 ∈푀 , 푎푥 denotesthe image of 푎 in 푀푥. For every 푅-module homomorphism 푓 : 푀 → 푁 ,the corresponding induced 푅푥-homomorphism 푓푥 : 푀푥 → 푁푥 is given by푓푥 = 푓 ⊗푅푥.Following [4] a ring 푅 is called uniform if for each 푥 ∈ 푆푝푒푐(퐵(푅))there exists a collection of isomorphisms (of rings) {휙푦 : 푅푦 → 푅푥∣푦 ∈푆푝푒푐(퐵(푅))} such that if 퐹 is a ﬁnite subset of 푅 there exists a neigh-borhood 푉 of 푥 with 휙푦(푎푦) = 푎푥 for all 푎 ∈ 퐹 and 푦 ∈ 푉 .The notion of uniform rings was generalized in [2]. A ring 푅 is calledlocally uniform if for each 푥 ∈ 푆푝푒푐(퐵(푅)) and each ﬁnite subset 퐹 ofJournal Algebra Discrete Math.22 A variant of the primitive element theorem푅 there exist a neighborhood 푈 = 푈(푥, 퐹 ) of 푥 and a collection of ringisomorphisms {휙푦 : 푅푦 → 푅푥∣푦 ∈ 푈} such that 휙푦(푎푦) = 푎푥, for every푎 ∈ 퐹 and 푦 ∈ 푈 . Consequently, if 푅 is locally uniform then there existan idempotent 푒 = 푒(푥, 퐹 ) ∈ 푅 and a collection of ring isomorphisms{휙푦 : 푅푦 → 푅푥∣푦 ∈ 푈푒} such that 푥 ∈ 푈푒 and 휙푦(푎푦) = 푎푥, for every푎 ∈ 퐹 and 푦 ∈ 푈푒.A ring 푅 is called von Neumann regular if for every element 푎 in 푅there exists an element 푏 in 푅 such that 푎 = 푎2푏, which is equivalent tosay that each element in 푅 is a product of an idempotent by a unit. Inparticular, von Neumann regular connected rings are ﬁelds.Examples of connected rings such that every boolean localizationmodulo its Jacobson radical is von Neumann regular and locally uni-form can be seen in [2]. In the sequel we present a ring with the aboveconditions that is not connected.Example. Let 푆 be a connected ring and 푅 =∏푛≥0 푆푛, where 푆푛 =푆 for all 푛 ≥ 0. Observe that the elements in 퐵(푅) are all of thetype (푎푛)푛∈ℕ with 푎푛 = 0 or 1. By [12, 2.2] one can easily see that푆푝푒푐(퐵(푅)) = {푥푖∣푖 ∈ ℕ}, where 푥푖 denotes the set of all elements(푎푛)푛∈ℕ ∈ 퐵(푅) such that 푎푖 = 0 and for every 푗 ∕= 푖 there exists anelement in 푥푖 whose 푗푡ℎ-coordinate is equal to 1. Consequently, 푅푥 ≃ 푆for all 푥 ∈ 푆푝푒푐(퐵(푅)), and in order to get the required it is enough totake 푆 such that 푆/퐽(푆) is von Neumann regular and locally uniform.2. Main resultA ring extension 푆 ⊃ 푅 is called separable if the multiplication map푚푆 : 푆 ⊗푅 푆 → 푆 is a splitting epimorphism of S-bimodules, which isequivalent to say that there exists an element 푥 ∈ 푆 ⊗푅 푆 which is 푆-central (i.e., 푥푠 = 푠푥 for all 푠 ∈ 푆) and satisﬁes the condition푚푆(푥) = 1푆 .Also, we say that 푆 is a strongly separable extension of푅 if 푆 is a separableextension of 푅 and 푆 is a ﬁnitely generated projective 푅-module.A polynomial 푓(푋) ∈ 푅[푋] is said to be separable over 푅 if it is monicand 푅[푋](푓(푋)) is a separable 푅-algebra. A monic polynomial 푓(푋) ∈ 푅[푋]is deﬁned to be indecomposable in 푅[푋] if whenever there exist monicpolynomials 푔(푋), ℎ(푋) ∈ 푅[푋] such that 푓(푋) = 푔(푋)ℎ(푋) it followsthat 푔(푋) = 1 or ℎ(푋) = 1.The purpose of this article is to prove the next theorem and its proofwill be divided in two parts: the connected and the general case.Theorem 2.1. Let 푅 be a ring such that 푅푥퐽(푅푥) is von Neumann regularand locally uniform, for every 푥 ∈ 푆푝푒푐(퐵(푅)), and 푆 a strongly separableJournal Algebra Discrete Math.D. Bagio, A. Paques 23extension of 푅. Then, there exist a strongly separable extension 푇 of 푅and an element 훼 in 푇 such that 푇 = 푅[훼] and 푆 ⊆ 푇 . If, in addition,푇 has constant rank over 푅, then there exists a separable polynomial푓(푋) ∈ 푅[푋] of degree 푟푎푛푘푅푇 such that 푓(훼) = 0 and 푇 ≃푅[푋](푓(푋)) .Connected caseTheorem 2.2. Let 푅 be a connected ring such that 푅퐽(푅) is von Neumannregular and locally uniform, and 푆 a strongly separable extension of 푅.Then, there exist a strongly separable extension 푇 of 푅, an element 훼 in푇 and a separable polynomial 푓(푋) ∈ 푅[푋] such that:(푖) 푆 ⊆ 푇 ;(푖푖) 푓(훼) = 0 and 푇 = 푅[훼] ≃ 푅[푋](푓(푋)) ;(푖푖푖) 퐵(푆) = 퐵(푇 ).Proof. This proof is quite similar to that of [2, Theorem 2.1]. Let 푅′ =푅퐽(푅) and 푆′ = 푆퐽(푆) . Note that 푅′푥 is a connected and von Neumannregular ring, so it follows that 푅′푥 is a ﬁeld, for all 푥 ∈ 푆푝푒푐(퐵(푅′)).Firstly assume that 푅′푥 is inﬁnite, for every 푥 ∈ 푆푝푒푐(퐵(푅′)). Thuseach 푆′푥 has a primitive element over 푅′푥 [6, Lemma 3.1] and, conse-quently, there exist 훼′(푥) ∈ 푆′ and an idempotent 푒(푥) ∈ 푅′ such that푥 ∈ 푈푒(푥) and 푆′푒(푥) = 푅′[훼′(푥)]푒(푥) [12, 2.8 and 2.11]. By compactnessarguments we obtain elements 훼′1, . . . , 훼′푛 ∈ 푆′ and orthogonal idem-potents 푒1, . . . , 푒푛 ∈ 푅′ such that∑1≤푖≤푛 푒푖 = 1 and 푆′푒푖 = 푅′[훼′푖]푒푖.Taking 훼′ =∑1≤푖≤푛 훼′푖푒푖 we have 푆′ = 푅′[훼′] and by Nakayama’s lemma푆 = 푅[훼] for some 훼 ∈ 푆 such that 훼′ = 훼+퐽(푆). Finally, (iii) is obviousand (ii) follows from [6, Theorem 2.9].Now put 푌 = {푥 ∈ 푆푝푒푐(퐵(푅′))∣푅′푥 is ﬁnite} and assume that 푌 ∕= ∅.By [11, Proposition 1.3] we can assume that 푆 = 푆1 ⊕ ⋅ ⋅ ⋅ ⊕ 푆푛, with 푆푖a connected and strongly separable extension of 푅. Moreover, by [6,Theorem 1.1] we may also assume that each 푆푖 is a connected Galoisextension of 푅 in the sense of [3].On the other hand, it follows from the proof of [2, Theorem 2.1]that for each 1 ≤ 푖 ≤ 푛 there exists a connected and strongly separableextension 푇푖 of 푅 such that 푆푖 ⊆ 푇푖 and 푟푎푛푘푆푖푇푖 = 푝푖, for some primeinteger 푝푖 satisfying푞푝푖−푞푝푖≥ 푟푎푛푘푅푆푖, where 푞 = 푚푖푛{∣푅′푥∣∣푥 ∈ 푌 }.Taking 푇 = 푇1 ⊕ ⋅ ⋅ ⋅ ⊕ 푇푛 we have that 푇 is a strongly separableextension of 푅, 푆 ⊆ 푇 , 퐵(푆) = 퐵(푇 ). It remains to show that 푇 alsosatisﬁes (ii).Put 푇 ′ = 푇퐽(푅)푇 =푇퐽(푇 ) . By boolean localization and Nakayama’slemma it is enough to prove that 푇 ′푥 has a primitive element over 푅′푥,Journal Algebra Discrete Math.24 A variant of the primitive element theoremfor every 푥 ∈ 푆푝푒푐(퐵(푅′)). If 푥 ∕∈ 푌 then 푇 ′푥 is an extension of 푅′푥 withprimitive element [6, Lemma 3.1].If 푥 ∈ 푌 , then 푅′푥 is a ﬁnite ﬁeld and 푅′푥 = 푅′/퐼(푥) with 퐼(푥) =픪/퐽(푅) for some maximal ideal 픪 of 푅. Again as in the proof of[2, Theorem 2.1] (see Claim 3), we have 푇푖픪푇푖≃ 푅/픪[푋](푓푖(푋)) , where each푓푖(푋) ∈ 푅/픪[푋] is separable over 푅/픪, of degree 푝푖(푟푎푛푘푅푆푖) and everyindecomposable factor of 푓푖(푋) in 푅/픪[푋] has the same degree 푝푖푑푖 with푑푖 a divisor of 푟푎푛푘푅푆푖.Furthermore, each 푝푖 can be chose such that 푝푖푑푖 ∕= 푝푗푑푗 if 푖 ∕= 푗.Therefore, the polynomials 푓푖(푋) are pairwise coprimes and푇픪푇 ≃푇1픪푇1⊕⋅ ⋅ ⋅⊕ 푇푛픪푇푛≃ 푅/픪[푋](푓(푋)) , with 푓(푋) =∏1≤푖≤푛 푓푖(푋). Since 푇′푥 = 푇′/퐼(푥)푇 ′ =푇/퐽(푇 )픪푇/퐽(푇 ) ≃ 푇/픪푇 , the proof is complete.The following example illustrates the type of construction consideredin the proof of Theorem 2.2.Example 2.3. Let 푅 = ℤ(2) be the localization of ℤ at 2ℤ and 푆 =푅⊕푅⊕푅. Clearly, 푆 is a strongly separable extension of 푅 and 푆 does nothave a primitive element over 푅. Take 픪 = 2푅, ℎ1(푋), ℎ2(푋), ℎ3(푋) ∈푅/픪[푋] separable and indecomposable polynomials with degrees 2, 3 and5 respectively, and 푓푖(푋) ∈ 푅[푋] monic polynomials such that ℎ푖(푋) =푓푖(푋) modulo 픪[푋], 1 ≤ 푖 ≤ 3. Taking 푇푖 =푅[푋](푓푖(푋))and 푇 = 푇1⊕푇2⊕푇3then 푇픪푇 ≃푅/픪[푋](ℎ(푋)) with ℎ(푋) = ℎ1(푋)ℎ2(푋)ℎ3(푋). Therefore, 푆 ⊆ 푇and 푇 is a strongly separable extension of 푅 with primitive element, byNakayama’s lemma.General caseIt is easy to check that a strongly separable extension 푆 ⊇ 푅 has aprimitive element if and only if 푆푥 ⊇ 푅푥 has a primitive element for all푥 ∈ 푆푝푒푐(퐵(푅)). A similar result is valid when we consider the variant(★) of the primitive element theorem.Lemma 2.4. Let 푅 be a ring. Then the following statements are equiva-lent:(푖) (★) is true for 푅.(푖푖) (★) is true for 푅푥, for all 푥 ∈ 푆푝푒푐(퐵(푅)).Proof. (i)⇒(ii) Let 푥 ∈ 푆푝푒푐(퐵(푅)) and 퐿 be a strongly separable exten-sion of 푅푥. By [7, II.24] there exists a strongly separable extension 푇 ofJournal Algebra Discrete Math.D. Bagio, A. Paques 25푅 such that 푇푥 = 퐿. Thus 푇 is contained in a strongly separable exten-sion 푇 ′ of 푅 having a primitive element, by assumption. Consequently푇 ′푥 is a strongly separable extension of 푅푥 having a primitive element and퐿 ⊆ 푇 ′푥.(ii)⇒(i) Let 푆 be a strongly separable extension of 푅 and assumethat for each 푥 ∈ 푆푝푒푐(퐵(푅)) there exists a strongly separable extension퐿(푥) of 푅푥, having a primitive element over 푅푥, such that 푆푥 ⊆ 퐿(푥).Then, there exist a strongly separable extension 푇(푥) of 푅 and an element훼(푥) ∈ 푇(푥) such that (푇(푥))푥 = 퐿(푥) = (푅[훼(푥)])푥 [7, II.24]. Consequently,there exists an idempotent 푒(푥) ∈ 푅 such that 푥 ∈ 푈푒(푥) and 푇(푥)푒(푥) =푅[훼(푥)]푒(푥) [12, 2.11].By compactness arguments we get pairwise orthogonal idempotents푒1, . . . , 푒푛 of푅, strongly separable extensions 푇1, . . . , 푇푛 of푅 and elements훼푖 ∈ 푇푖 such that∑1≤푖≤푛 푒푖 = 1 and 푇푖푒푖 = 푅[훼푖]푒푖, 1 ≤ 푖 ≤ 푛. Taking푇 = 푇1푒1 ⊕ ⋅ ⋅ ⋅ ⊕ 푇푛푒푛 and 훼 = 훼1푒1 + ⋅ ⋅ ⋅ + 훼푛푒푛 we have that 푇 is astrongly separable extension of 푅 and 푇 = 푅[훼].It remains to prove that 푆 ⊆ 푇 . Take 푠 ∈ 푆 and 푥 ∈ 푆푝푒푐(퐵(푅)).By construction 푆푥 ⊆ 푇푥, so there exists 푡 ∈ 푇 such that 푠푥 = 푡푥 andconsequently 푠− 푡 ∈ 퐼(푥)푇 ⊆ 푇 .Corollary 2.5. Let 푅 be a ring. If 푅푥퐽(푅푥) is von Neumann regular andlocally uniform for any 푥 ∈ 푆푝푒푐(퐵(푅)), then (★) is true for 푅.Proof. It follows from Theorem 2.2 and Lemma 2.4.Corollary 2.6. If the prime spectrum 푆푝푒푐(푅) of a ring 푅 is totallydisconnected then (★) is true for 푅.Proof. Indeed, in this case 푅푥 is semilocal for all 푥 ∈ 푆푝푒푐(퐵(푅)) [5].Then the result follows by Corollary 2.5.Now we are able to prove Theorem 2.1.Proof of Theorem 2.1. The ﬁrst assertion follows by Corollary2.5. For the second assume that 푇 = 푅[훼] has constant rank 푛 over푅. Then 푇푥 = (푅[훼])푥 = 푅푥[훼푥] and by [6, Theorem 2.9] there exists amonic polynomial 푓(푥)(푋) in 푅[푋] of degree 푛, such that (푓(푥)(푋))푥 isseparable over 푅푥 and (푓(푥)(훼))푥 = 0 for each 푥 ∈ 푆푝푒푐(퐵(푅)).The separability of (푓(푥)(푋))푥 implies that (휆(푥)푑(푓(푥)(푋)))푥 = 1푥 forsome 휆(푥) ∈ 푅, where 푑(푓(푥)(푋)) denotes the discriminant of 푓(푥)(푋).By [12, 2.9] there exists an idempotent 푒(푥) ∈ 푅 such that 푥 ∈ 푈푒(푥),푓(푥)(훼)푒(푥) = 0 and (휆(푥)푑(푓(푥)(푋)))푒(푥) = 푒(푥), which means that푓(푥)(푋)푒(푥) is separable over 푅푒(푥).Journal Algebra Discrete Math.26 A variant of the primitive element theoremBy compactness arguments we get orthogonal idempotents 푒1, . . . , 푒푚of 푅 and monic polynomials 푓1(푋), . . . , 푓푚(푋) ∈ 푅[푋] of degree 푛 suchthat∑1≤푖≤푚 푒푖 = 1, 푓푖(푋)푒푖 is separable over 푅푒푖 and 푓푖(훼)푒푖 = 0.Put 푓(푋) =∑1≤푖≤푚 푓푖(푋)푒푖. Thus, 푓(푋) is a polynomial of degree 푛,separable over 푅 and 푓(훼) = 0.Finally the canonical map 휑 : 푅[푋] → 푅[훼], ℎ(푋) 7→ ℎ(훼), is anepimorphism of 푅-algebras whose kernel contains 푓(푋). Hence, it inducesan epimorphism from 푅[푋]/(푓(푥)) onto 푅[훼]. Since 푟푎푛푘푅푅[훼] = 푛 =푟푎푛푘푅(푅[푋]/(푓(푋))) it follows that 휑 is an isomorphism.References[1] D. Bagio, I. Dias and A. Paques, On self-dual normal bases, Indag. Math. 17(2006), 1-11.[2] D. Bagio and A. Paques, A generalized primitive element theorem, Math. J.Okayama Univ. 49 (2007), 171-181.[3] S.U. Chase, D.K. Harrison and A. Rosenberg, Galois theory and Galois cohomol-ogy of commutative rings, Mem. Amer. Math. Soc. 52 (1968), 1-19.[4] F. DeMeyer, Separable polynomials over a commutative ring, Rocky Mountain J.of Math. 2 (1972), 299-310.[5] F. DeMeyer, On separable polynomials over a commutative ring, Pacific J. Math.51 (1974), 57-66.[6] G. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc.122 (1966) 461-479.[7] A.R. Magid, The Separable Galois Theory of Commutative Rings, Marcel Dekker,NY, 1974.[8] T. McKenzie, The separable closure of a local ring, J. of Algebra 207 (1998),657-663.[9] A. Paques, On the primitive element and normal basis theorems, Comm. in Al-gebra 16 (1988), 443-455.[10] J-D. Therond, Le the´ore`me de l’e´le´ment primitif pour un anneau semilocal, J. ofAlgebra 105 (1987), 29-39.[11] O.E Villamayor and D. Zelinsky, Galois theory with finitely many idempotents,Nagoya Math. J. 27 (1966), 721-731.[12] , Galois theory with infinitely many idempotents, Nagoya Math. J. 35(1969), 83-98.Contact informationD. Bagio Departamento de Matema´ticaUniversidade Federal de Santa Maria97105-900, Santa Maria, RS, BrazilE-Mail: bagio@smail.ufsm.brJournal Algebra Discrete Math.D. Bagio, A. Paques 27A. Paques Instituto de Matema´ticaUniversidade Federal do Rio Grande do Sul91509-900, Porto Alegre, RS, BrazilE-Mail: paques@mat.ufrgs.brReceived by the editors: 12.08.2009and in ﬁnal form 25.09.2009.

A variant of the primitive element theorem for separable extensions of a commutative ring

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A variant of the primitive element theorem for separable extensions of a commutative ring

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