We show, how Gröbner bases can be computed for two-sided ideals of iterated Ore extensions with commuting variables.
Given a ring R an iterated Ore extension with commuting variables. Identifying the iterated Ore extension of R and the polynomial ring over R (in the same number of variables) as free left R-Modules all two sided ideals of the iterated Ore extension are left ideals of the polynomial ring.
We therefore define a Gröbner basis of a two-sided ideal seen as a left ideal of the corresponding polynomial ring. This, of course, requires that left Gröbner bases exist in the polynomial ring.
If there is an algorithm for computing a left Gröbner basis for any given finite subset of the polynomial ring this algorithm can be extended to compute two-sided Gröbner bases in the iterated Ore extension.
Examples of ground rings R meeting this requirement are polynomial rings over fields or over PID's and solvable polynomial rings.
Ulrike Peiker, Martin Griebl