On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

关于除环的伪理想

M.K.Sen于1976年推广环的理想概念,定义了伪理想,并用依理想来研究除环、域和正则环。侯国荣1983年又推广伪理想,建立n—伪理想,得到了更好的结果。本文则证明了:非交换除环和有限域无真伪理想,因而文关于除环的几乎全部结果都是平凡的。我们还证明了非交换除环无真n—伪理想,给出了有限域无真n—伪理想的充分必要条件。文中讨论了无限域的伪理想,得到了关于除环结构的几个有用结果。     

On division subrings normalized by almost subnormal subgroups in division rings

Periodica Mathematica Hungarica - Let D be a division ring with infinite center, K a proper division subring of D and N an almost subnormal subgroup of the multiplicative group $$D^*$$ of D. The...     

On derived groups of division rings II

Let D be a division ring with centre F and denote by D′ the derived group (commutator subgroup) of D ^? = D ? {0}. It is shown that if each element of D′ is algebraic over F, then D is algebraic over F. It is also proved that each finite separable extension of F in D is of the form F(c) for some element c in the derived group D′. Using these results, it is shown that if each element of the derived group D′ is of bounded degree over F, then D is finite dimensional over F.     

On division rings generated by polycyclic groups

LetD=F(G) be a division ring generated as a division ring by its central subfieldF and the polycyclic-by-finite subgroupG of its multiplicative group, letn be a positive integer and letX be a finitely generated subgroup of GL(n, D). It is implicit in recent works of A. I. Lichtman thatX is residually finite. In fact, much more is true. If charD=p≠0, then there is a normal subgroup ofX of finite index that is residually a finitep-group. If charD=0, then there exists a cofinite set π=π(X) of rational primes such that for eachp in π there is a normal subgroup ofX of finite index that is residually a finitep-group.     

On anti-automorphisms of the first kind in division rings

We provide a short proof of the well-known fact that a division ring of finite degree over its center that admits an anti-automorphism of the first kind, i.e., an anti-automorphism that fixes the center elementwise, also admits an involutive anti-automorphism.

     

On schur rings of group rings of finite groups

Schur rings are rings associated to certain partitions of finite groups. They were introduced for applications in representation theory, cfr. [3][4].

The algebric structure of these rings has not been studied in depth. In this paper we determine explicit structure constants for Schur rings, we derive conditions for separability and we compute the centre. These results seem to be new even over fields.     

On Lie metabelian group rings

Let G be a finite group without elements of orders two and three and R be a commutative ring with characteristic different from 2. If either the subrings A of R(G), the group ring of G over R, generated by the set {g + g^?1; g ∈ G} or B generated by the set {g ? g^?1; g ∈ G} is Lie metabelian, then G is abelian.     

On the multiplicative structure of finite division rings

All division ringsD of 16, 27, 32, 125 and 343 elements are shown to have aright primitive elementp such that

On group rings of nilpotent groups

It is shown that ifG is a non-abelian torsion free nilpotent group andF is a field, then the classical skew field of fractionsF(G) of the group ring,F[G] contains a noncommutative free subalgebra. The author is supported by NSF Grant No. MCS-8201115.     

On group rings of linear groups

Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number $N=N(H)$. We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field.     

On locally finite alternative division rings with valuation

We prove that a locally finite complete alternative division ring with valuation is a field, without using the celebrated theorem of Bruck and Kleinfeld, which classifies all division rings with IP.The author's research is supported by the National Fund for Scientific research (Belgium).