Let D be a division ring with centre F. An element of the form xyx −1 y −1 ∈ D is called a 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.
We characterise finite unitary rings
$R$
such that all Sylow subgroups of the group of units
$R^{\ast }$
are cyclic. To be precise, we show that, up to isomorphism,
$R$
is one of the three types of rings in
$\{O,E,O\oplus E\}$
, where
$O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$
is a ring of odd cardinality and
$E$
is a ring of cardinality
$2^{n}$
which is one of seven explicitly described types.
Cartan–Brauer–Hua Theorem is a well-known theorem which states that if [Formula: see text] is a subdivision ring of a division ring [Formula: see text] which is invariant under all elements of [Formula: see text] or [Formula: see text] for all [Formula: see text], then either [Formula: see text] or [Formula: see text] is contained in the center of [Formula: see text]. The invariance idea of this basic theorem is the main notion of this paper. We prove that if [Formula: see text] is a division ring with involution [Formula: see text] and [Formula: see text] is a subspace of [Formula: see text] which is invariant under all symmetric elements of [Formula: see text], then either [Formula: see text] is contained in the center of [Formula: see text] or is a Lie ideal of [Formula: see text]. Also, we show that if [Formula: see text] is a self-invariant subfield of a non-commutative division ring [Formula: see text] with a nontrivial automorphism, then [Formula: see text] contains at least one non-central proper subfield of [Formula: see text].
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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