On Noncommutative Involutive Divisions

Buchberger’s algorithm for computing a Gröbner basis solves the ideal membership problem over commutative polynomial rings. In the early 1990’s, an alternative to this algorithm was found, namely the involutive basis algorithm, which provides a Gröbner basis with extra combinatorial properties. Buchberger’s work was generalised to noncommutative polynomial rings by Bergman and Mora during the 1970’s and 1980’s. This article provides the corresponding generalisation for involutive bases, providing a noncommutative involutive basis algorithm, and analysing several noncommutative involutive divisions for use with the algorithm.