Bui Xuan Hai, Trinh Thanh Deo, Mai Hoang Bien
Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.
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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 aim of this paper is to show that if $K$ is $N$-invariant and $N$ is non-central, then $K$ is central. Some examples of almost subnormal subgroups in division rings that are not subnormal are also given.
We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in [3] and of Herstein in [6].
In this paper we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains free non-cyclic subgroups.
Let $D$ be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^*$ of $D$. The new obtained results generalize some previous results on subgroups of $D^*$.
Let $D$ be a division ring with center $F$ and $K$ a division subring of $D$. In this paper, we show that a non-central normal subgroup $N$ of the multiplicative group $D^*$ is left algebraic over $K$ if and only if so is $D$ provided $F$ is uncountable and contained in $K$. Also, if $K$ is a field and the $n$-th derived subgroup $D^{(n)}$ of $D^{*}$ is left algebraic of bounded degree $d$ over $K$, then $\dim_FD\le d^2$.
Let $D$ be a division ring with center $F$, and $G$ a subnormal or quasinormal subgroup of $D^*$. We show that if $G$ is locally solvable, then $G$ is contained in $F$.
We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders.
Let $D$ be a division ring and $D^*$ be the multiplicative group of $D$. In this paper we study locally solvable maximal subgroups of $D^*$.
The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.
Let $D$ be a division ring with center $F$, and $G$ a subnormal subgroup of $D^*$. We show that if $G$ is a locally solvable group such that $G^{(i)}$ is algebraic over $F$, then $G$ must be central. Also, if $M$ is non-abelian locally solvable maximal subgroup of $G$ with $M^{(i)}$ algebraic over $F$, then $D$ is a cyclic algebra of prime degree over $F$.
In this paper we study non-central almost subnormal subgroups of the multiplicative group of a division ring satisfying a non-zero generalized rational identity. The main result generalizes Chiba's theorem on subnormal subgroups. As an application, we get a theorem on almost subnormal subgroups satisfying a generalized algebraic rational identity. The last theorem has several corollaries which generalize completely or partially some previous results.
Let $K$ be a field and let $σ$ be an automorphism and let $δ$ be a $σ$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;σ,δ)$ contains a free non-cyclic subgroup unless $D$ is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.
In this note, we introduce a new concept of a {\it generalized algebraic rational identity} to investigate the structure of division rings. The main theorem asserts that if a non-central subnormal subgroup $N$ of the multiplicative group $D^*$ of a division ring $D$ with center $F$ satisfies a non-trivial generalized algebraic rational identity of bounded degree, then $D$ is a finite dimensional vector space over $F$. This generalizes some previous results.
In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.
Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center by locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occu…
Let E be a graded division ring finite-dimensional over its center with torsion-free abelian grade group, and let q(E) be its quotient division ring. Let tau be a degree-preserving unitary involution on E. We prove that SK_1(E, tau) is isomorphic to SK_1(q(E), tau).
We develop the theory of group graded division ring parallel to the one by P. Cohn for (ungraded) division rings.
The Central Nullstellensatz over Centrally Algebraically Closed Division Rings
本文引入中心代数封闭除环概念，证明除环满足中心 Nullstellensatz 当且仅当其中心代数封闭，且任何除环均可嵌入此类除环。
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
Weakly locally finite division rings were considered in \cite{dbh}, where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer $n\geq 0$ or $n=\infty$, we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is $n$. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well…
For a division ring D, finite dimensional over its center F, we give a condiction for the connectedness of the commuting graph of a matrix ring over $D$. Furthermore, we prove that if the commuting graph is connected, then its diameter is between four and six.
We give examples of locally PI but not PI division rings of GK-dimension n for every positive integer n.
We give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its centre, thus disproving the conjecture posed in the paper "Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8".
Roots of polynomials over division rings
研究除环上多项式的性质，并给出求某些多项式根的公式。
In this paper, we give the definition of {\em weakly locally finite} division rings and we show that the class of these rings strictly contains the class of locally finite division rings. Further, we study multiplicative subgroups in these rings. Some skew linear groups are also considered. Our new obtained results generalize previous results for centrally finite case.
It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.
Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.
Polynomials over division rings
该文研究除环上多项式的性质，得到任意除环上多项式的分解定理，证明非中心非根元素的共轭类含无穷多非根元素，还估计了四元数可除代数上几类多项式球面根的共轭类数量。
Torsion subgroups of Whitehead groups of graded division algebras
研究有限维分次除环\(E\)的怀特黑德群的挠子群\({\rm TK}_1(E)\)，给出特殊情形下\({\rm TK}_1(E)\)公式，借此计算有限维赋值除环\(D\)的\({\rm TK}_1(D)\)，为相关研究提供了计算方法。
We first give a sufficient condition for a Mal'cev-Neumann ring of formal series to be a noncrossed product division algebra. This result is then used to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p^2 for p any odd prime).
Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.
The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.
Commutator products in skew Laurent series division rings
针对非交换除环元素能否表示为两个换位子乘积的问题，本文对域上的斜洛朗级数除环给出了肯定答案，推进了单环与除环结构相关研究。
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.
Let $D$ be a division ring infinite-dimensional over its center $k$ with multiplicative group $D^{\times}$. We show that if $D$ belongs to certain families, there exist free symmetric and unitary pairs in $D^{\times}$ with respect to a $k$-involution on $D$.
Let $D$ be a division ring, $n$ a positive integer, and GL$_n(D)$ the general linear group of degree $n$ over $D$. In this paper, we study the induced subgraph of the intersection graph of GL$_n(D)$ generated by all non-trivial proper almost subnormal subgroups of GL$_n(D)$. We show that this subgraph is complete if it is non-null. This property will be used to study subgroup structure of a division ring. In particular, we prove that every non-central almost subnormal subgroup of the multiplicative group $D^*$ of a division ring $D$ contains a non-central subnormal subgroup of $D^*$.
Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e. ND*(K)= K: This kind of subfields is important because they have strong connexion with the most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dime…
Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a certain set of generators in the induced graded ring $grad(R)$ implies the existence of free group algebras in $R$. Our best results are obtained for division rings endowed with a valuation.
Makar-Limanov's conjecture states that if a division ring D is finitely generated and infinite dimensional over its center k then D contains a free k-subalgebra of rank 2. In this work, we will investigate the existence of such structures in D, the division ring of fractions of the skew polynomial ring L[t;σ], where t is a variable and $σ$ is a k-automorphism of L. For instance, we prove Makar-Limanov's conjecture when either L is the function field of an abelian variety or the function field of the n-dimensional projective space.
Exploring Functional Identities: From Division Rings to Matrix Algebras
本文通过严格分析与定理证明，解决了Ferreira等人提出的未决问题，深入探究了除环上矩阵代数中加性函数的复杂行为，并揭示了其对代数结构的影响。
We prove that the fixed ring of the $q$-division ring $k_q(x,y)$ under any finite group of monomial automorphisms is isomorphic to $k_q(x,y)$ for the same $q$. In a similar manner, we also show that this phenomenon extends to an automorphism that is defined only on $k_q(x,y)$ and does not restrict to $k_q[x^{\pm1},y^{\pm1}]$. We then use these results to answer several questions posed by Artamonov and Cohn about the endomorphisms and automorphisms of $k_q(x,y)$.
Deformed Laurent series rings and completions of the Weyl division ring
该研究定义了形变洛朗级数环并给出其充要条件，还构造了特征0域上Weyl除环$D_1$关于一类离散赋值的完备化，证明其具有赋值环为非交换亨泽尔环等良好性质。
We show that the canonical involution on a nonabelian poly-orderable group G extends to the Hughes-free division ring of fractions D of the group algebra k[G] of G over a field k and that, with respect to this involution, D contains a pair of symmetric elements freely generating a free group subalgebra of D over k.
A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K. Special cases of such polynomials include, for instance, the minimal polynomials (over the center F=Z(K)) of elements of K that are algebraic over F. In this note, we give a survey on some of our ongoing work on the structure theory of Wedderburn polynomials. Throughout the note, we work in the general setting of an Ore skew polynomial ring K[t,S,D].
In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.
Let $k$ be an algebraically closed field of characteristic zero, let $X$ and $Y$ be smooth irreducible algebraic curves over $k$, and let $D(X)$ and $D(Y)$ denote respectively the quotient division rings of the ring of differential operators of $X$ and $Y$. We show that if there is a $k$-algebra embedding of $D(X)$ into $D(Y)$ then the genus of $X$ must be less than or equal to the genus of $Y$, answering a question of the first-named author and Smoktunowicz.
Study of a division-like property
本文为非交换环引入弱类除性性质（fadelian），证明其性质并构造非除环、非Noetherian及非Ore的例子，且在Lean中形式化部分结果。
Let T be a complete discrete valuation ring and $\hat{X}$ a smooth projective curve over $S=\spec(T)$ with closed fibre $X$. Denote by $F$ the function field of $\hat{X}$ and by $\hat{F}$ the completion of $F$ with respect to the discrete valuation defined by $X$, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over $F$. This is done by defining an index preserving group homomorphism $s:\br(\hat{F})'\to\br(F)'$, and using it to lift indecomposable and noncrossed product division algebras over $\hat{F}$.
Let $\mathbb{F}$ be a division ring. In this paper, we extent some of the main well-known results about the resultant of two univariate polynomials to the more general context of an Ore extension $\mathbb{F}[x;σ,δ]$. Finally, some algorithms and Magma programs are given as a numerical application of the main theoretical results of this paper.
As is well known, any complex cyclic matrix $A$ is similar to the unique companion matrix associated with the minimal polynomial of $A$. On the other hand, a cyclic matrix over a division ring $\mathbb F$ is similar to a companion matrix of a polynomial which is defined up to polynomial similarity. In this paper we study more rigid canonical forms by embedding a given cyclic matrix over a division ring $\mathbb F$ into a controllable or an observable pair. Using the characterization of ideals in $\mathbb F[z]$ in terms of controllable and observable pairs we consider ideal interpolation scheme…
Finite groups that are embeddable in the multiplicative groups of division rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic groups, according to a theorem of I. N. Herstein. Thus, the only interesting case is when $K$ has characteristic 0; that is, when $K\supseteq {\Bbb Q}$. Herstein conjectured that odd-order subgroups of division rings $K$ were cyclic, and he proved this to be the case when $K$ is the division ring of the real quaternions. Herstein's conjecture was settled negatively b…
In this note, the correction to the proof of one theorem in some our previous paper [arXiv:1302.0589] will be given.
Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a in D^(n); the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.
A system of linear equations over a skew field has properties similar to properties of a system of linear equations over a field. Even noncommutativity of a product creates a new picture the properties of system of linear equations and of vector space over skew-field have a close relationship.
A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in "Wedderburn polynomials over division rings,I (Journal of Pure and Applied Algebra, Vol. 186, (2004), 43-76). In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of $(S,D)$-pseudo-linear transformations. In the last section we introduce and study the notion o…
Let $D$ be a division ring of fractions of a crossed product $F[G,η,α]$ where $F$ is a skew field and $G$ is a group with Conradian left-order $\leq$. For $D$ we introduce the notion of freeness with respect to $\leq$ and show that $D$ is free in this sense if and only if $D$ can canonically be embedded into the endomorphism ring of the right $F$-vector space $F((G))$ of all formal power series in $G$ over $F$ with respect to $\leq$. From this we obtain that all division rings of fractions of $F[G,η,α]$ which are free with respect to at least one Conradian left-order of $G$ are isomorphic and …
A celebrated theorem of P.M.Cohn says that for any two division rings (not necessarily finite dimensional) over a field F, their amalgamated product over F is a domain which can be embedded in a division ring. Note that even with the two initial division rings begin finite dimensional over their centers, the resulting division ring is never finite dimensional over its center. Perhaps this led Lance Small to ask the following question. Assume $F_1$ and $F_2$ are fields with the same characteristic. Small asked whether any two division algebras $D_1/F_1$ and $D_2/F_2$ can be embedded in some thi…
We apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings. If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_L$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_L$ generated by $U(L)$. Let $k$ be a field of characteristic zero and $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group a…
A note on division rings satisfying generalized rational identities with anti-automorphisms
本文研究带非平凡反自同构幂 $σ^m$ 的除环，证明其若满足相应广义有理恒等式，则必为中心有限除环。
Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the $\mathbb{F}$-vector space $M_{n,p}(\mathbb{D})$ in which every matrix has rank less than or equal to $r$. Using a new method, we prove that $\dim_{\mathbb{F}} \mathcal{V} \leq \max(n,p)\,rd$ and we characterize the spaces for which equality holds. This extends a famous theorem of Flanders which was known only for fields.
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