arXiv subjects

Hau-Yuan Jang

Publications and source records attributed to Hau-Yuan Jang.

3 recordsLinked to original sources

arXiv2026

A Reduced-Trace-Zero Element That Is Not a Commutator in a Central Division Algebra

We exhibit a finite-dimensional central division algebra, of degree four over its center, together with an explicit element of reduced trace zero that is not a single commutator. This answers negatively the single-commutator question for finite-dimensional central division algebras, the case left open by Amitsur and Rowen. The algebra is a skew Laurent series ring, the element has only two terms, and the proof uses the $x$-adic valuation, a first-obstruction lemma based on two commuting involutions of the associated graded ring, and a parity argument for quadratic forms over iterated Laurent series fields.

math.RA↗

arXiv2025

Products of commutators in simple algebras

Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as $a=(bc-cb)(de-ed)$ for some $b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb N$ we provide an example of an infinite-dimensional simple unital $C^*$-algebra $A$ in which $1$ cannot be written as $\sum_{i=1}^m x_i(a_ib_i-b_ia_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$.

math.RA↗

arXiv2025

Commutator products in skew Laurent series division rings

In 1965, Baxter established that a simple ring is either a field or that every one of its elements can be expressed as a sum of products of commutator pairs. In a recent paper, Gardella and Thiel demonstrated that every element in a noncommutative division ring can be represented as the sum of just two products of two commutators. They further posed the question of whether every element in a noncommutative division ring can be represented as the product of two commutators. In this paper, we affirmatively answer this question for skew Laurent series division rings over fields.

math.RA↗