On Additive Commutator Groups in Division Rings

Akbari, S.; Ariannejad, M.; Mehrabadi, M. L.

doi:10.1007/bf03322065

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…generality, assume that a␣ / ␣ a. Assume also that A, B g GL L are 2 2 k the matrix representations of a, ␣ , respectively. It is clear that for each…”

Section: Conditions For Commutativitymentioning

confidence: 99%

Maximal Subgroups of GL1(D)

1999

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Let D be a division algebra of degree m over its center F. Herstein has shown Ž . that any finite normal subgroup of D* [ GL D is central. Here, as a generaliza-1 tion of this result, it is shown that any finitely generated normal subgroup of D* is Ž central. This also solves a problem raised by Akbari and Mahdavi-Hezavehi Proc.. Amer. Math. Soc., to appear for finite-dimensional division algebras. The structure of maximal multiplicative subgroups of an arbitrary division ring D is then investigated. Given a maximal subgroup M of D* whose center is algebraic over F, it is proved that if M satisfies a multilinear polynomial identity over F, then w x D : F -ϱ. ᮊ 1999 Academic Press

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“…generality, assume that a␣ / ␣ a. Assume also that A, B g GL L are 2 2 k the matrix representations of a, ␣ , respectively. It is clear that for each…”

Section: Conditions For Commutativitymentioning

confidence: 99%

Maximal Subgroups of GL1(D)

1999

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“…It is clear that for each x ∈ Q, the matrix representation of x + α 2 is B x = xI + B. Since N is finitely generated, by the argument used in the proof of Theorem 3, we conclude that there is a set Λ = {m 1 …”

Section: This Contradicts the Noetherian Conditionmentioning

confidence: 90%

“…Also, in Chapter 3 of [4], there are several group theoretic conditions whose occurrence in the multiplicative group of a division algebra entails the commutativity of the algebra. Much research in recent years has been focused on the study of the structure of the multiplicative subgroups of division algebras; for example see [1], [3], [6], [7], [8], [10], [11], and [12] for an introduction. Before stating our results, we fix some notation.…”

Section: Introductionmentioning

confidence: 99%

Normal subgroups of $GL_n(D)$ are not finitely generated

Akbari

Mahdavi-Hezavehi

1999

Proc. Amer. Math. Soc.

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“…Additive commutator elements of a ring R and the groups and structures they make have a great role in the general specification of a ring, and their study is one of the approaches to recognize rings in noncommutative ring theory [2,3,4,5]. The reason is clear, they have covered the secrets of noncommutative behaviour of the structure.…”

Section: Introductionmentioning

confidence: 99%

Notes on Whitehead space of an algebra

Ariannejad

2002

International Journal of Mathematics and Mathematical Sciences

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LetRbe a ring, and denote by[R,R]the group generated additively by the additive commutators ofR. WhenRn=Mn(R)(the ring ofn×nmatrices overR), it is shown that[Rn,Rn]is the kernel of the regular trace function modulo[R,R]. Then consideringRas a simple left ArtinianF-central algebra which is algebraic overFwithChar F=0, it is shown thatRcan decompose over[R,R], asR=Fx+[R,R], for a fixed elementx∈R. The spaceR/[R,R]overFis known as the Whitehead space ofR. WhenRis a semisimple centralF-algebra, the dimension of its Whitehead space reveals the number of simple components ofR. More precisely, we show that whenRis algebraic overFandChar F=0, then the number of simple components ofRis greater than or equal todimF R/[R,R], and whenRis finite dimensional overFor is locally finite overFin the case ofChar F=0, then the number of simple components ofRis equal todimF R/[R,R].

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On Additive Commutator Groups in Division Rings

Cited by 11 publications

References 9 publications

Maximal Subgroups of GL1(D)

Maximal Subgroups of GL1(D)

Normal subgroups of $GL_n(D)$ are not finitely generated

Notes on Whitehead space of an algebra

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