Narrow Results

Let $D$ be a division ring, $\sigma$ an automorphism of $D$ and $u\in D$ a fixed element. We determine the forms of additive maps $f,g:D\to D$ satisfying the identity

Given a finite dimensional F-central simple algebra A = M_n(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GL_n(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

Let D be an F-central non-commutative division ring. Here, it is proved that if GL_n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL_n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL₂(F) for fields F with Char F ≠ 2.

Let R be a ring. If we replace the original associative product of R with their canonic Lie product, or [a, b] = ab - ba for every a, b in R, then R would be a Lie ring. With this new product the additive commutator subgroup of R or [R, R] is a Lie subring of R. Herstein has shown that in a simple ring R with characteristic unequal to 2, any Lie ideal of R either is contained in Z(R), the center of R or contains [R, R]. He also showed that in this situation the Lie ring [R, R]/Z[R, R] is simple. We give an alternative matrix proof of these results for the special case of simple artinian rings and show that in this case the characteristic condition can be more restricted.

Let D be a division ring with center F. An element of the form xyx^-1y^-1 ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char(D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F.

Let R be an associative ring with identity, and let I be an (left, right, two-sided) ideal of R. Say that I is small if |I| < |R| and large if |R/I| < |R|. In this paper, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of R. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals.

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if $R$ is a subdivision ring of a division ring $D$ which is invariant under all elements of $D$ or $dR{d}^{-1}\subseteq R$ for all $d\in D\setminus \{0\}$, then either $R=D$ or $R$ is contained in the center of $D$. The invariance idea of this basic theorem is the main notion of this paper. We prove that if $D$ is a division ring with involution ${}^{\ast }$ and $M$ is a subspace of $D$ which is invariant under all symmetric elements of $D$, then either $M$ is contained in the center of $D$ or is a Lie ideal of $D$. Also, we show that if $T$ is a self-invariant subfield of a non-commutative division ring $D$ with a nontrivial automorphism, then $T$ contains at least one non-central proper subfield of $D$.

We show that if $I$ is a non-central Lie ideal of a ring $R$ with Char$(R)\ne 2$, such that all of its nonzero elements are invertible, then $R$ is a division ring. We prove that if $R$ is an $F$-central algebra and $I$ is a Lie ideal without zero divisor such that the set of multiplicative cosets $\{aF|a\in I\}$ is of finite cardinality, then either $R$ is a field or $I$ is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central $F$-algebra $R$ with Char$(R)\ne 2$ and radical over the center is $[R,R]$, the additive commutator subgroup of $R$ and in this case $R$ is a generalized quaternion algebra. Finally we prove that if $I$ is a Lie ideal without zero divisor in a central $F$-algebra with characteristic not 2 and if $(\frac{I+F}{F},+)$ is a finite residual group, then $I$ is central.

We show that there is no nontrivial group homomorphism $E_n(R)\to E_{n-1}(R)$ over commutative local rings and division rings for $n\ge 3$, respectively. It gives a negative answer to Ye’s problem (see [S. K. Ye, Low-dimensional representations of matrix group actions on CAT(0) spaces and manifolds, J. Algebra409 (2014) 219–243]) for the above rings.

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 $l\in L,a\in A$ is in $L$ or $[l,a]\in L$. The main concern of this paper is to present some properties of Lie ideals of $A$ which may be interpreted as being dual to known properties of normal subgroups of $A^{\ast }$. In particular, we prove that if $A$ is a finite-dimensional division algebra with center $F$ and $\mathrm{\text{char}}F\ne 2$, then any finitely generated $\mathbb{Z}$-module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A,A]$ of $A$ is not a finitely generated $\mathbb{Z}$-module. Some other results about maximal additive subgroups of $A$ and $M_n(A)$ are also presented.

We give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its center, thus disproving the conjecture posed in [M. Aghabali, S. Akbari, M. Ariannejad and A. Madadi, Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl.12(8) (2013) 7 pp.].

In this paper, we characterize Lie ideals, which are either finitely generated $\mathbb{Z}$-modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not $2$ to simple rings of arbitrary characteristic.

Let $R$ be a noncommutative division ring with center $Z$, which is algebraic, that is, $R$ is an algebraic algebra over the field $Z$. Let $f$ be an antiautomorphism of $R$ such that (i) ${[f(x),x^{m(x)}]}_{n(x)}=0$, all $x\in R$, where $m(x)$ and $n(x)$ are positive integers depending on $x$. If, further, $f$ has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that $f$ is commuting, that is, $[f(x),x]=0$, all $x\in R$. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on $f$ can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring $R$ with an antiautomorphism $f$ satisfying the stronger condition (ii) ${[f(x),x^m]}_n=0$, all $x\in R$, where $m$ and $n$ are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, $f$ has finite order then $f$ is commuting. We show here, that again the finite order assumption on $f$ can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

In this paper, we study some algebras $A$ whose unit groups ${𝒰}(A)$ or subnormal subgroups of ${𝒰}(A)$ are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.

Given a non-commutative finite-dimensional $F$-central division ring $D$, $N$ a subnormal subgroup of ${\mathrm{\text{GL}}}_n(D)$ and $M$ a non-abelian maximal subgroup of $N$, then either $M$ contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup $A$ of $M$ such that $K=F[A]$ is a subfield of $M_n(D)$, $K/F$ is Galois and $\mathrm{\text{Gal}}(K/F)\cong M/(K^{\ast }\cap N)$, also $\mathrm{\text{Gal}}(K/F)$ is a finite simple group with $F[M]=M_n(D)$.

In this paper, we study non-central almost subnormal subgroups of the multiplicative group of a division ring satisfying a nonzero 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 $F$ be a real-closed field and $D$ a $F$-division algebra. In this paper, we prove that if $D$ is algebraic over $F$ then the graph $\mathrm{\Gamma }({\mathrm{\text{M}}}_n(D))$ is connected and its diameter is at most $4$, for any $n\ge 3$. If in addition, the division ring $D$ is noncommutative, we also have the same results for $n=2$. As a corollary, we show that the diameter of the commuting graph of the matrix algebra of degree $n\ge 2$ over a generalized quaternion algebra ${ℍ}_F(a,b)$, where $F$ is a real-closed field, two elements $a,b\in F^{\ast }$, is also at most $4$. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most $6$ in case $F$ is the field $ℝ$ of real numbers.

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GL_n(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GL_n(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A₁X = C₁, XB₂ = C₂, A₃XB₃ = C₃ and A₄XB₄ = C₄ over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.