Noncommutative Symmetric Algebras of Two-Sided Vector Spaces

We generalize the definition of the symmetric algebra of a vector space in order to define noncommutative symmetric algebras of two-sided vector spaces. Not all two-sided vector spaces have noncommutative symmetric algebras; the ones that do are called admissible, and conditions for admissibility are given. Further, for some classes of admissible two-sided vector spaces, the skew fields of fractions of their noncommutative symmetric algebras are computed. The degree 0 components of these skew fields correspond to function fields of certain noncommutative ruled surfaces, and hence allow us to determine birational equivalence classes for such surfaces.