Preface Note to the reader Terminology, notations and conventions used List of special notation 0. Preliminaries on modules 1. Principal ideal domains 2. Firs, semifirs and the weak algorithm 3. Factorization 4. 2-firs with a distributive factor lattice 5. Modules over firs and semifirs 6. Centralizers and subalgebras 7. Skew fields of fractions Appendix Bibliography and author index Subject index.

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Free Ideal Rings and Localization in General Rings

Let D be a division ring with center k, and let D † be its multiplicative group. We investigate the existence of free groups in D †, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.

https://bv.fapesp.br/pt/publicacao/pdf/56463/a-survey-on-free-objects-in-division-rings-and-in-division-r

A Survey on Free Objects in Division Rings and in Division Rings with an Involution

Let $D$ be a weakly locally finite division ring and $n$ a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group ${\rm GL}_n(D)$. Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of ${\rm GL}_n(D)$ are described.

Free subgroups in almost subnormal subgroups of general skew linear groups

Let $K$ be a field, let $\sigma$ be an automorphism of $K$, and let $\delta$ be a derivation of $K$. We show that if $D$ is one of $K(x;\sigma)$ or $K(x;\delta)$, then $D$ either contains a free algebra over its center on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we are able to show that the quotient division ring of any iterated Ore extension of an affine domain satisfying a polynomial identity either again satisfies a polynomial identity or it contains a free algebra over its center on two variables.

/pdf/free-subalgebras-of-quotient-rings-of-ore-extensions-4f8b4l7pzf.pdf

Free subalgebras of quotient rings of Ore extensions

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

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Katsuo Chiba

Papers

Free Subgroups and Free Subsemigroups of Division Rings

Free fields in complete skew fields and their valuations