Noncommutative symmetric algebras of two-sided vector spaces (Q5925791)

Let \(k\) be a field, and let \(E\) be a free \(k\)-bimodule, or in other words, a two-sided \(k\)-vector space. The author introduces a concept of admissible two-sided \(k\)-vector spaces and defines for such a space a noncommutative analog of the symmetric algebra. Then he computes the skew fields of fractions for some classes of admissible two-sided vector spaces. The components of degree zero of these skew fields can be considered as function fields of noncommutative ruled surfaces (cf. [\textit{M. Artin, M. Van den Bergh}, J. Algebra 133, No. 2, 249-271 (1990; Zbl 0717.14001)]). As an application a clear description of the birational equivalence classes of certain types of such surfaces is obtained.