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 space s. Not all two-sided vector spaces have noncommutative symmetric algebras; the ones that do are called admissible, and conditions for admissibility ar e given. Further, for some classes of admissible two-sided vector spaces, t he skew fields of fractions of their noncommutative symmetric algebras are computed. The degree 0 components of these skew fields correspond to functi on fields of certain noncommutative ruled surfaces, and hence allow us to d etermine birational equivalence classes fur such surfaces. (C) 2000 Academi c Press.