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

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

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.

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Datum:2009
Hauptverfasser: Bagio, D., Paques, A.
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Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2009
Schriftenreihe:Algebra and Discrete Mathematics
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spelling irk-123456789-1546182019-06-16T01:27:50Z A variant of the primitive element theorem for separable extensions of a commutative ring Bagio, D. Paques, A. 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. 2009 Article A variant of the primitive element theorem for separable extensions of a commutative ring / D. Bagio, A. Paques // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 20–26. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:13B05, 12F10 http://dspace.nbuv.gov.ua/handle/123456789/154618 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description 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.
format Article
author Bagio, D.
Paques, A.
spellingShingle Bagio, D.
Paques, A.
A variant of the primitive element theorem for separable extensions of a commutative ring
Algebra and Discrete Mathematics
author_facet Bagio, D.
Paques, A.
author_sort Bagio, D.
title A variant of the primitive element theorem for separable extensions of a commutative ring
title_short A variant of the primitive element theorem for separable extensions of a commutative ring
title_full A variant of the primitive element theorem for separable extensions of a commutative ring
title_fullStr A variant of the primitive element theorem for separable extensions of a commutative ring
title_full_unstemmed A variant of the primitive element theorem for separable extensions of a commutative ring
title_sort variant of the primitive element theorem for separable extensions of a commutative ring
publisher Інститут прикладної математики і механіки НАН України
publishDate 2009
url http://dspace.nbuv.gov.ua/handle/123456789/154618
citation_txt A variant of the primitive element theorem for separable extensions of a commutative ring / D. Bagio, A. Paques // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 20–26. — Бібліогр.: 12 назв. — англ.
series Algebra and Discrete Mathematics
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2009). pp. 20 – 26 c⃝ Journal “Algebra and Discrete Mathematics” A variant of the primitive element theorem for separable extensions of a commutative ring Dirceu Bagio and Antonio Paques Communicated by guest editors Abstract. In this article we show that any strongly sep- arable extension of a commutative ring R can be embedded into another one having primitive element whenever every boolean lo- calization of R modulo its Jacobson radical is von Neumann regular and locally uniform. Dedicated to Professor Miguel Ferrero on occasion of his 70-th anniversary Introduction Throughout this paper by ring we mean a commutative ring with identity element. By a connected ring we mean a ring whose unique idempotents are 0 and 1. Furthermore, J(R) denotes the Jacobson radical of the ring R. Given a ring extension S ⊇ R we say that S has a primitive element over R if there exists � ∈ S such that S = R[�]. As it is well known, any finite separable field extension has a primitive element. Although this assertion is not true in general, several authors has been obtained extensions of it for strongly separable extensions S of a ring R. For instance, in the case that: — (R,m) is a local ring with ∣ ∣ R m ∣ ∣ = ∞ [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. Jo u rn al A lg eb ra D is cr et e M at h .D. Bagio, A. Paques 21 — R is a semilocal ring with ∣ ∣ R m ∣ ∣ ≥ rankRS, for every maximal ideal m of R [10]; — R is a ring with many units and such that ∣ ∣ R m ∣ ∣ ≥ rankRS, for every maximal ideal m of R [9]. An alternative and also interesting question is to know under what conditions the following variant of the primitive element theorem holds: (★) any strongly separable extension S of a ring R can be embedded into another one having primitive element. The statement (★) is true for connected strongly separable extensions of R in the following cases: R is local [8, Theorem 1.1], R is connected and semilocal [1, Theorem 2.1.1] and R is connected with R J(R) von Neumann regular and locally uniform [2, Theorem 2.1]. In this paper we show that the statement (★) is valid for any ring R, provided that Rx J(Rx) is von Neumann regular and locally uniform, for all prime ideal x in the boolean ring B(R) of all idempotents of R. 1. Preliminaries In this paper we will be employing freely the ideas and results of [12] on boolean spectrum and boolean localization of a ring (see also [7]). We begin by recalling the terminology we will need. For any ring R let B(R) denote the boolean ring of all idempotents of R and Spec(B(R)) the boolean spectrum of R consisting of all prime (equivalently maximal) ideals of B(R) (see [12, 2.1 and 2.2]). A base for a topology on Spec(B(R)) is given by the family of basic open sets {Ue∣e ∈ B(R)}, where Ue = {x ∈ Spec(B(R))∣1 − e ∈ x}. This base defines a compact, totally disconnected, Hausdorff topology on Spec(B(R)). By localization of R at x, for each x ∈ Spec(B(R)), we mean the quotient ring Rx = R I(x) where I(x) denotes the ideal of R generated by the elements of x. By [12, 2.13] Rx is a connected ring. For any R-module M , we set Mx = M ⊗R Rx = M I(x)M . For any element a ∈ M , ax denotes the image of a in Mx. For every R-module homomorphism f : M → N , the corresponding induced Rx-homomorphism fx : Mx → Nx is given by fx = f ⊗Rx. Following [4] a ring R is called uniform if for each x ∈ Spec(B(R)) there exists a collection of isomorphisms (of rings) {�y : Ry → Rx∣y ∈ Spec(B(R))} such that if F is a finite subset of R there exists a neigh- borhood V of x with �y(ay) = ax for all a ∈ F and y ∈ V . The notion of uniform rings was generalized in [2]. A ring R is called locally uniform if for each x ∈ Spec(B(R)) and each finite subset F of Jo u rn al A lg eb ra D is cr et e M at h .22 A variant of the primitive element theorem R there exist a neighborhood U = U(x, F ) of x and a collection of ring isomorphisms {�y : Ry → Rx∣y ∈ U} such that �y(ay) = ax, for every a ∈ F and y ∈ U . Consequently, if R is locally uniform then there exist an idempotent e = e(x, F ) ∈ R and a collection of ring isomorphisms {�y : Ry → Rx∣y ∈ Ue} such that x ∈ Ue and �y(ay) = ax, for every a ∈ F and y ∈ Ue. A ring R is called von Neumann regular if for every element a in R there exists an element b in R such that a = a2b, which is equivalent to say that each element in R is a product of an idempotent by a unit. In particular, von Neumann regular connected rings are fields. Examples of connected rings such that every boolean localization modulo 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 above conditions that is not connected. Example. Let S be a connected ring and R = ∏ n≥0 Sn, where Sn = S for all n ≥ 0. Observe that the elements in B(R) are all of the type (an)n∈ℕ with an = 0 or 1. By [12, 2.2] one can easily see that Spec(B(R)) = {xi∣i ∈ ℕ}, where xi denotes the set of all elements (an)n∈ℕ ∈ B(R) such that ai = 0 and for every j ∕= i there exists an element in xi whose jtℎ-coordinate is equal to 1. Consequently, Rx ≃ S for all x ∈ Spec(B(R)), and in order to get the required it is enough to take S such that S/J(S) is von Neumann regular and locally uniform. 2. Main result A ring extension S ⊃ R is called separable if the multiplication map mS : S ⊗R S → S is a splitting epimorphism of S-bimodules, which is equivalent to say that there exists an element x ∈ S ⊗R S which is S- central (i.e., xs = sx for all s ∈ S) and satisfies the condition mS(x) = 1S . Also, we say that S is a strongly separable extension of R if S is a separable extension of R and S is a finitely generated projective R-module. A polynomial f(X) ∈ R[X] is said to be separable over R if it is monic and R[X] (f(X)) is a separable R-algebra. A monic polynomial f(X) ∈ R[X] is defined to be indecomposable in R[X] if whenever there exist monic polynomials g(X), ℎ(X) ∈ R[X] such that f(X) = g(X)ℎ(X) it follows that g(X) = 1 or ℎ(X) = 1. The purpose of this article is to prove the next theorem and its proof will be divided in two parts: the connected and the general case. Theorem 2.1. Let R be a ring such that Rx J(Rx) is von Neumann regular and locally uniform, for every x ∈ Spec(B(R)), and S a strongly separable Jo u rn al A lg eb ra D is cr et e M at h .D. Bagio, A. Paques 23 extension of R. Then, there exist a strongly separable extension T of R and an element � in T such that T = R[�] and S ⊆ T . If, in addition, T has constant rank over R, then there exists a separable polynomial f(X) ∈ R[X] of degree rankRT such that f(�) = 0 and T ≃ R[X] (f(X)) . Connected case Theorem 2.2. Let R be a connected ring such that R J(R) is von Neumann regular and locally uniform, and S a strongly separable extension of R. Then, there exist a strongly separable extension T of R, an element � in T and a separable polynomial f(X) ∈ R[X] such that: (i) S ⊆ T ; (ii) f(�) = 0 and T = R[�] ≃ R[X] (f(X)) ; (iii) B(S) = B(T ). Proof. This proof is quite similar to that of [2, Theorem 2.1]. Let R′ = R J(R) and S′ = S J(S) . Note that R′ x is a connected and von Neumann regular ring, so it follows that R′ x is a field, for all x ∈ Spec(B(R′)). Firstly assume that R′ x is infinite, for every x ∈ Spec(B(R′)). Thus each S′ x has a primitive element over R′ x [6, Lemma 3.1] and, conse- quently, there exist �′(x) ∈ S′ and an idempotent e(x) ∈ R′ such that x ∈ Ue(x) and S′e(x) = R′[�′(x)]e(x) [12, 2.8 and 2.11]. By compactness arguments we obtain elements �′ 1, . . . , � ′ n ∈ S′ and orthogonal idem- potents e1, . . . , en ∈ R′ such that ∑ 1≤i≤n ei = 1 and S′ei = R′[�′ i]ei. Taking �′ = ∑ 1≤i≤n � ′ iei we have S′ = R′[�′] and by Nakayama’s lemma S = R[�] for some � ∈ S such that �′ = �+J(S). Finally, (iii) is obvious and (ii) follows from [6, Theorem 2.9]. Now put Y = {x ∈ Spec(B(R′))∣R′ x is finite} and assume that Y ∕= ∅. By [11, Proposition 1.3] we can assume that S = S1 ⊕ ⋅ ⋅ ⋅ ⊕ Sn, with Si a connected and strongly separable extension of R. Moreover, by [6, Theorem 1.1] we may also assume that each Si is a connected Galois extension of R in the sense of [3]. On the other hand, it follows from the proof of [2, Theorem 2.1] that for each 1 ≤ i ≤ n there exists a connected and strongly separable extension Ti of R such that Si ⊆ Ti and rankSi Ti = pi, for some prime integer pi satisfying qpi−q pi ≥ rankRSi, where q = min{∣R′ x∣∣x ∈ Y }. Taking T = T1 ⊕ ⋅ ⋅ ⋅ ⊕ Tn we have that T is a strongly separable extension of R, S ⊆ T , B(S) = B(T ). It remains to show that T also satisfies (ii). Put T ′ = T J(R)T = T J(T ) . By boolean localization and Nakayama’s lemma it is enough to prove that T ′ x has a primitive element over R′ x, Jo u rn al A lg eb ra D is cr et e M at h .24 A variant of the primitive element theorem for every x ∈ Spec(B(R′)). If x ∕∈ Y then T ′ x is an extension of R′ x with primitive element [6, Lemma 3.1]. If x ∈ Y , then R′ x is a finite field and R′ x = R′/I(x) with I(x) = m/J(R) for some maximal ideal m of R. Again as in the proof of [2, Theorem 2.1] (see Claim 3), we have Ti mTi ≃ R/m[X] (fi(X)) , where each fi(X) ∈ R/m[X] is separable over R/m, of degree pi(rankRSi) and every indecomposable factor of fi(X) in R/m[X] has the same degree pidi with di a divisor of rankRSi. Furthermore, each pi can be chose such that pidi ∕= pjdj if i ∕= j. Therefore, the polynomials fi(X) are pairwise coprimes and T mT ≃ T1 mT1 ⊕ ⋅ ⋅ ⋅⊕ Tn mTn ≃ R/m[X] (f(X)) , with f(X) = ∏ 1≤i≤n fi(X). Since T ′ x = T ′/I(x)T ′ = T/J(T ) mT/J(T ) ≃ T/mT , the proof is complete. The following example illustrates the type of construction considered in the proof of Theorem 2.2. Example 2.3. Let R = ℤ(2) be the localization of ℤ at 2ℤ and S = R⊕R⊕R. Clearly, S is a strongly separable extension of R and S does not have a primitive element over R. Take m = 2R, ℎ1(X), ℎ2(X), ℎ3(X) ∈ R/m[X] separable and indecomposable polynomials with degrees 2, 3 and 5 respectively, and fi(X) ∈ R[X] monic polynomials such that ℎi(X) = fi(X) modulo m[X], 1 ≤ i ≤ 3. Taking Ti = R[X] (fi(X)) and T = T1⊕T2⊕T3 then T mT ≃ R/m[X] (ℎ(X)) with ℎ(X) = ℎ1(X)ℎ2(X)ℎ3(X). Therefore, S ⊆ T and T is a strongly separable extension of R with primitive element, by Nakayama’s lemma. General case It is easy to check that a strongly separable extension S ⊇ R has a primitive element if and only if Sx ⊇ Rx has a primitive element for all x ∈ Spec(B(R)). A similar result is valid when we consider the variant (★) of the primitive element theorem. Lemma 2.4. Let R be a ring. Then the following statements are equiva- lent: (i) (★) is true for R. (ii) (★) is true for Rx, for all x ∈ Spec(B(R)). Proof. (i)⇒(ii) Let x ∈ Spec(B(R)) and L be a strongly separable exten- sion of Rx. By [7, II.24] there exists a strongly separable extension T of Jo u rn al A lg eb ra D is cr et e M at h .D. Bagio, A. Paques 25 R such that Tx = L. Thus T is contained in a strongly separable exten- sion T ′ of R having a primitive element, by assumption. Consequently T ′ x is a strongly separable extension of Rx having a primitive element and L ⊆ T ′ x. (ii)⇒(i) Let S be a strongly separable extension of R and assume that for each x ∈ Spec(B(R)) there exists a strongly separable extension L(x) of Rx, having a primitive element over Rx, such that Sx ⊆ L(x). Then, there exist a strongly separable extension T(x) of R and an element �(x) ∈ T(x) such that (T(x))x = L(x) = (R[�(x)])x [7, II.24]. Consequently, there exists an idempotent e(x) ∈ R such that x ∈ Ue(x) and T(x)e(x) = R[�(x)]e(x) [12, 2.11]. By compactness arguments we get pairwise orthogonal idempotents e1, . . . , en of R, strongly separable extensions T1, . . . , Tn of R and elements �i ∈ Ti such that ∑ 1≤i≤n ei = 1 and Tiei = R[�i]ei, 1 ≤ i ≤ n. Taking T = T1e1 ⊕ ⋅ ⋅ ⋅ ⊕ Tnen and � = �1e1 + ⋅ ⋅ ⋅ + �nen we have that T is a strongly separable extension of R and T = R[�]. It remains to prove that S ⊆ T . Take s ∈ S and x ∈ Spec(B(R)). By construction Sx ⊆ Tx, so there exists t ∈ T such that sx = tx and consequently s− t ∈ I(x)T ⊆ T . Corollary 2.5. Let R be a ring. If Rx J(Rx) is von Neumann regular and locally uniform for any x ∈ Spec(B(R)), then (★) is true for R. Proof. It follows from Theorem 2.2 and Lemma 2.4. Corollary 2.6. If the prime spectrum Spec(R) of a ring R is totally disconnected then (★) is true for R. Proof. Indeed, in this case Rx is semilocal for all x ∈ Spec(B(R)) [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 Corollary 2.5. For the second assume that T = R[�] has constant rank n over R. Then Tx = (R[�])x = Rx[�x] and by [6, Theorem 2.9] there exists a monic polynomial f(x)(X) in R[X] of degree n, such that (f(x)(X))x is separable over Rx and (f(x)(�))x = 0 for each x ∈ Spec(B(R)). The separability of (f(x)(X))x implies that (�(x)d(f(x)(X)))x = 1x for some �(x) ∈ R, where d(f(x)(X)) denotes the discriminant of f(x)(X). By [12, 2.9] there exists an idempotent e(x) ∈ R such that x ∈ Ue(x), f(x)(�)e(x) = 0 and (�(x)d(f(x)(X)))e(x) = e(x), which means that f(x)(X)e(x) is separable over Re(x). Jo u rn al A lg eb ra D is cr et e M at h .26 A variant of the primitive element theorem By compactness arguments we get orthogonal idempotents e1, . . . , em of R and monic polynomials f1(X), . . . , fm(X) ∈ R[X] of degree n such that ∑ 1≤i≤m ei = 1, fi(X)ei is separable over Rei and fi(�)ei = 0. Put f(X) = ∑ 1≤i≤m fi(X)ei. Thus, f(X) is a polynomial of degree n, separable over R and f(�) = 0. Finally the canonical map ' : R[X] → R[�], ℎ(X) 7→ ℎ(�), is an epimorphism of R-algebras whose kernel contains f(X). Hence, it induces an epimorphism from R[X]/(f(x)) onto R[�]. Since rankRR[�] = n = rankR(R[X]/(f(X))) it follows that ' 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 théorème de l’élément primitif pour un anneau semilocal, J. of Algebra 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 information D. Bagio Departamento de Matemática Universidade Federal de Santa Maria 97105-900, Santa Maria, RS, Brazil E-Mail: bagio@smail.ufsm.br Jo u rn al A lg eb ra D is cr et e M at h .D. Bagio, A. Paques 27 A. Paques Instituto de Matemática Universidade Federal do Rio Grande do Sul 91509-900, Porto Alegre, RS, Brazil E-Mail: paques@mat.ufrgs.br Received by the editors: 12.08.2009 and in final form 25.09.2009.

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