Let D be a division ring with centre F. Let T(D) be the vector space over F
generated by all multiplicative commutators in D. In [1], authors have
conjectured that every division ring is generated as a vector space over its
centre by all of its multiplicative commutators. In this note it is shown that
if D is centrally finite, then the conjecture holds

Aaghabali, Mehdi

Tajfirouz, Zakeieh

English

arXiv

Let D be a division ring with centre F. An element of the form xyx−1y−1∈ D is calleda multiplicative commutator. Let T (D) be the vector space over F generated by all multiplicative commutators in D. In [1], authors have conjectured that every division ring is generated as a vector space over its centre by all of its multiplicative commutators. In this note it is shown that if D is centrally finite, then the conjecture holds

Aghabali, Mehdi

Tajfirouz, Z.

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Remarks on Vector Space Generated by the MultiplicativeCommutators of a Division RingCitation for published version:Aghabali, M & Tajfirouz, Z 2017 'Remarks on Vector Space Generated by the Multiplicative Commutators ofa Division Ring' ArXiv. <https://arxiv.org/abs/1707.00658>Link:Link to publication record in Edinburgh Research ExplorerGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 01. Nov. 2022M. Aaghabali a, M. Ariannejad b, A. Madadi caSchool of Mathematics, The University of Edinburgh, James Clerk Maxwell Building,The Kings Buildings, Mayfield Road EH9 3JZ, Edinburgh, UKbDepartment of Mathematics, University of Zanjan, Zanjan, IrancDepartment of Mathematics, Zanjan Branch Azad Islamic University, Zanjan, IranAbstract A Lie ideal of a division ring A is an additive subgroup L of A such that the Lie product[l, a] = la − al of any two elements l ∈ L, a ∈ A is in L, or [l, a] ∈ L. The main concern of this paper is topresent some properties of Lie ideals of A which may be interpreted as being dual to known properties ofnormal subgroups of A∗. In particular, we prove that if A is a finite-dimensional division algebra with centerF and charF 6= 2, then any finitely generated Z-module Lie ideal of A is central. We also show that theadditive commutator subgroup [A,A] of A is not a finitely generated Z-module. Some other results aboutmaximal additive subgroups of A and Mn(A) are also presented.AMS Classification: 17A01, 17A35.Keywords: Lie ideal, division ring, finitely generated.1 PreliminaryLet A be an associative algebra over a field F with dimension [A : F ]. If we replace the usual multiplicationab of two elements a and b of A by their Lie product [a, b] = ab − ba, then at the same time we have a non-associative structure of a Lie ring on A, usually denoted by A. A Lie ideal in A is a regular ideal of A with itsLie multiplication; in other words, an additive subgroup I of A is called a Lie ideal if for all i ∈ I, a ∈ A wehave [i, a] ∈ I. The main concern of this paper is to present some dual properties of these two structures onA: considering A as an associative algebra and as a Lie ring. These properties represent very similar roles ofnormal subgroups in A∗ (the multiplicative group of unit elements) and (Lie) ideals in A. In particular, when Ais a division ring we give some properties of Lie ideals of A which are analogous to similar properties of normalsubgroups of A∗.A derivation on A is an additive group homomorphism d : A → A satisfying d(ab) = d(a)b + ad(b). Aderivation da : A → A that is defined by da(b) = ab − ba, for some fixed a ∈ A is called inner derivation; theset {da|a ∈ A} of all inner derivations of A is denoted by InnDer(A). In group theory, the normal subgroupsof a group G usually are defined as the subgroups which are invariant under all inner automorphisms of G(denoted by Inn(G)). Equivalently, in the theory of Lie algebras, Lie ideals of an algebra A are defined as thesubmodules which are invariant under all inner derivations of A. Let Z(A) denote the center of A; then we havethe following similar isomorphisms: Inn(A∗) ' A∗/Z(A∗) and as a dual version InnDer(A) ' A/Z(A) [10, p.73].The following two main theorems give some more important signs in identifying a connection between theconcepts of normal subgroups and Lie ideals: Let F be a field, then the SkolemNoether theorem (in particular)states that if A is a finite-dimensional central simple F -algebra, then any F -automorphism of A is inner [3, p.93]. A dual version of this theorem states that if A is a finite-dimensional central simple F -algebra, then anyF -linear derivation of A is inner [3, p. 105].The other one is the CartanBrauerHua theorem which states that if A is a division ring and B is a subdivisionring of A such that B∗ is a normal subgroup of A∗, then either B = A or B ⊆ Z(A) [6, p. 211]. A dual versionof this theorem states that if A is a division ring and B is a subdivision ring of A such that B is a Lie ideal inA and charA 6= 2, then either B = A or B ⊆ Z(A) [6, p. 205].We consider some results about the structure of normal subgroups of a division ring and examine theirdual versions in terms of Lie ideals of the division ring. As an example, Akbari et al. [1, 2] proved that “IfE-mail Addresses: mehdi.aaghabali@ed.ac.uk, arian@znu.ac.ir, as.madadi@yahoo.comThe authors are indebted to the referee for careful reading and detailed valuable comments, which improved the paper. Theresearch of the first author was supported with ERC [grant number 320974].1A is a finite-dimensional division algebra with center F, then any finitely generated normal subgroup of A∗is central”. Also, they proved that “If A is an infinite division ring with center F such that [A : F ] < ∞,then A∗ contains no finitely generated maximal subgroups”. Here, as an analogous statement, we show that“If A is a finite-dimensional division algebra with center F such that charF 6= 2, then any finitely generatedZ-module Lie ideal of A is central”. We also show that “If A is an infinite division ring with center F such that[A : F ] < ∞, then A contains no finitely generated Z-module maximal additive subgroup”. We show that theadditive commutator subgroup [A,A] of A is not a finitely generated Z-module. To sum up, the applicability ofsimilar arguments we used to prove these dual properties reveals similar roles of these two substructures: thenormal subgroups and Lie ideals in division rings [7, 8, 9].2 Main ResultsWe begin by recalling the following theorem:Theorem 1.[4, p. 5] Let A be a division algebra with center F and char(A) 6= 2. Assume that L is a Lie idealof A. Then either L ⊆ F or [A,A] ⊆ L.This result allows us to present our first main result:Theorem 2. Let A be a division ring which is finite-dimensional over its center F and charF 6= 2. If A containsa non-central Lie ideal which is a finitely generated Z-module, then F is finitely generated over its prime subfieldP.Proof. Let L be a non-central finitely generated Z-module Lie ideal of A. By Theorem 1, [A,A] ⊆ L. Let T bethe F -subdivision algebra generated by L. Since any non-commutative division ring is generated as a divisionring by all of its additive commutators together with its center [6, p. 205], we conclude that T = A. Note thatsince A is a finite-dimensional division ring, L generates A as an algebra, too. If [A : F ] = n, then A has afaithful matrix representation θ of degree n [5, p. 82] (usually called the regular representation). Since L is afinitely generated Z-module, there exist a finite number k of matrices M1, . . . ,Mk in GLn(F ) which generateθ(L) in Mn(F ) as a Z-module. Let Γ ⊆ F be the set of elements of F that appear as entries in the matricesM1, . . . ,Mk. Since L builds A as an algebra, invoking θ one can see that this set of matrices first builds θ(L)and then builds A ' θ(A) in Mn(F ). Since θ is an embedding, we may consider A = T ⊆Mn(P (γ)) ⊆Mn(F ),where P (Γ) is the subfield of F generated by P ∪ Γ. Consequently, for all a ∈ F, its representation aI is inMn(P (Γ)), where I is the identity matrix and so a ∈ P (Γ) or F = P (Γ). 2We need the following lemma to present our next results.Lemma 3. Let D be a UFD with infinitely many prime ideals and let T be its field of fractions. Let A be aT -subalgebra of Mn(T ). Then any Lie ideal of A which is finitely generated as a Z-module is central.Proof. For the sake of contradiction, assume that L is a non-central Lie ideal of A which is finitely generatedas a Z-module. Let a ∈ A and l ∈ L be such that [a, l] 6= 0. Since L is finitely generated as a Z-module, there isa nonzero d ∈ D such that L ⊆Mn(D[ 1d ]). Since T is a field, for any x ∈ T we have x[a, l] = [xa, l] ∈ L. Hencex[a, l] ∈ Mn(D[ 1d ]). Since [a, l] ∈ Mn(T ) is a nonzero matrix, one of its entries is nonzero, say b. Therefore,xb ∈ D[ 1d ] for all x ∈ T, which is a contradiction, for if p is a prime element such that p - d, then b/pn /∈ D[ 1d ]for enough large positive integer n. 2Theorem 4. Let A be a finite-dimensional division algebra with center F such that charF 6= 2. Then anyfinitely generated Z-module Lie ideal of A is central.Proof. By Theorem 2, F is finitely generated over its prime subfield P. Hence we may write F as a finiteextension of a purely transcendental extension P (x1, . . . , xd) of P, where d is the transcendence degree of Fover P. We consider two cases:Case 1. d = 0. If P = Fp, then F is a finite field. Hence by the Wedderburns Little Theorem, A iscommutative [6, p. 203]. If P = Q, then [F : Q] < ∞ allows us to view A ∈ Mn(Q) via the regularrepresentation. Now, using the above lemma, we are done.Case 2. d > 0. Then P [x1, . . . , xd] is a UFD with infinitely many prime ideals. Let T be the field of fractionsof D. Since [F : T ] < ∞, again we may view A ∈ Mn(T ) via the regular representation. Now, applying theabove Lemma completes the proof 2The following is our main result:Corollary 5. Let A be a non-commutative division algebra of finite dimension over its center F and charF 6= 2.Then the additive commutator subgroup [A,A] of A is not finitely generated as a Z-module.2The Lie ideal structure we have considered above really is a kind of additive subgroup of algebras. In whatfollows, we turn our attention to another kind of additive subgroups. By a maximal additive subgroup of analgebra, we mean an additive subgroup which is maximal under inclusion among proper ones. Clearly, by amaximal Lie ideal, we mean a Lie ideal which is maximal under inclusion among Lie ideals.Corollary 6. Let A be a division ring with center F and charF 6= 2. Assume that L is a proper maximaladditive subgroup of A containing F. If the additive group index [L : F ] of L over F is finite, then A = F.Proof. First, consider the case [A : F ] < ∞ and let x1, . . . , xt be the representations of the finite number ofcosets of F in L, so L = (F + x1) ∪ · · · ∪ (F + xt). We have L = F + 〈{x1, . . . , xt}〉, where 〈{x1, . . . , xt}〉 isthe additive subgroup generated by x1, . . . , xt in A. Suppose that x ∈ A \ L. By maximality of L, we obtainthat A = F + 〈{x1, . . . , xt, x}〉. Put H = 〈{x1, . . . , xt, x}〉. Thus, A = F + H and consequently [A,A] =[H,H]. This means that [H,H] is a Lie ideal of A which is finitely generated as a Z-module by the finite set{xixj − xjxi, xxi − xix; i, j = 1, . . . , t}. By Theorem 4, we conclude that [A,A] = [H,H] ⊆ F or A = F asdesired. Now, consider the case [A : F ] =∞. As in the above case, let L = (F + x1) ∪ · · · ∪ (F + xt) and takex ∈ A \ L. Let V be the vector space generated by the set {1, x1, . . . , xt, x} over F. Clearly [V : F ] < ∞ andL $ V. Now, maximality of L implies that V = A, a contradiction. This completes the proof. 2We continue our study with the following two lemmas:Lemma 7. Let A be an F -algebra and L be a maximal Lie ideal of A. Then(i) L contains either F or [A,A].(ii) If A is a division ring, then either A = F (L) or L \ {0} is the multiplicative group F (L) \ {0}, where F (L)is the division ring generated by F ∪ L.Proof. (i) Assume that L does not contain F. By maximality of L and since F + L is a Lie ideal containingL, we have A = F + L. Consequently, we have [A,A] = [L,L] ⊆ L.(ii) Consider the division ring F (L) generated by L and F. By maximality of L and since F (L) is a Lie idealcontaining L, we have either A = F (L) or L = F (L). In the latter case, we obtain F (L)∗ = F (L)\{0} = L\{0}is a multiplicative group. 2Lemma 8. Let A be a division ring with center F and assume that L is a maximal Lie ideal of A. Then eitherthe multiplicative center of L is equal to F ∩ L or L is a maximal division subring of A.Proof. By first part of the previous lemma, either F ⊆ L or [A,A] ⊆ L. If [A,A] ⊆ L, then Z(L) = CL(L) ⊆CA(L) ⊆ CA([A,A]) = F, where the latter inclusion is by [6, p. 205]. In other words, Z(L) ⊆ F ∩ L and soZ(L) = F ∩L. If F ⊆ L and [A,A] is not contained in L, then consider the division ring F (L). Since F (L) is aLie ideal containing L, we have either A = F (L) or L = F (L) by the maximality of L in A. In the first case, itis easily checked that Z(L) = Z(A) = F ∩ L. Otherwise, L = F (L) which means that L is a maximal divisionsubring of A. 2Now, we can show that Theorem 4 has an analogous statement which applies to maximal additive subgroups.Theorem 9. Let A be a non-commutative division ring with center F. Then A contains no finitely generatedZ-module maximal additive subgroup.Proof. Assume that L is a maximal additive subgroup of A that is finitely generated as a Z-module. Foreach element x ∈ A \ L, we have A = L+ Zx which means that A is a finitely generated Z-module and this isimpossible: If charF 6= 0, then A would be finite and so commutative. If charF = 0, this condition makes Ato be finite-dimensional over the center. So A is a finitely generated Z-module Lie ideal of a finite-dimensionaldivision ring A which by Theorem 4 is contained in F and thus is commutative, contradicting our assumptionthat A is non-commutative. 2Theorem 10. Let A be a division algebra algebraic over its center F with charF 6= 2 and let n be a naturalnumber. Assume that L is a maximal Lie ideal of Mn(A). If L is finite, then A = F.Proof. Let I ∈ Mn(A) be the identity matrix. By Lemma 7(i), either FI ⊆ L or [Mn(A),Mn(A)] ⊆ L. Thelatter case implies that [A,A] is finite, so A = F by Corollary 6. If FI ⊆ L, then F is finite and so A a divisionalgebra algebraic over a finite field would be commutative and thus A = F. 23References[1] S. Akbrai and M. Mahdavi-Hezavehi, Normal subgroups of GLn(D) are not finitely generated, Proc. Amer.Math. Soc. 128(6) (2000) 16271632.[2] S. Akbari, M. Mahdavi-Hezavehi and M. G. Mahmudi, Maximal subgroups of GL1(D), J. Algebra 217(1999) 422433.[3] B. Farb and R. K. Dennis, Noncommutative Algebra, Graduate Text in Mathematics, Vol. 144 (Springer-Verlag, 1993).[4] I. N. Herstein, Topics in Ring Theory (The University of Chicago Press, 1972).[5] R. S. Pierce, Associative Algebras (Springer-Verlag, New York, 1982).[6] T. Y. Lam, A First Course in Noncommutative Rings, 2nd edn., Graduate Text in Mathematics, Vol. 131(Springer-Verlag, New York, 2001).[7] M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of GL1(D), J. Algebra 241 (2001) 720730.[8] M. Mahdavi-Hezavehi and S. Akbari, Some special subgroups of GLn(D), Algebra Colloq. 5(4) (1998)361370.[9] M. Mahdavi-Hezavehi, M. G. Mahmudi and S. Yasamin, Finitely generated subnormal subgroups ofGLn(D) are central, J. Algebra 225 (2000) 517521.[10] L. H. Rowen, Ring Theory, Student edition (Academic Press, Boston, 1991).4

Remarks on Vector Space Generated by the Multiplicative Commutators of a Division Ring

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