Abstract
It is shown that finite Gröbner bases exist and can be computed for two-sided ideals of iterated Ore extensions (which are also called iterated skew polynomial rings) with commuting variables.
Given a ring R consider an iterated Ore extension of R where the new variables commute with each other.
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 of the iterated Ore extension as a Gröbner basis of this two-sided ideal regarded 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 and solvable polynomial rings over fields or over principal ideal domains.
Applications include solving the two-sided ideal membership problem and computing in residue class rings of two sided ideals.
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Pesch, M. (1998). Two-sided Gröbner Bases in Iterated Ore Extensions. In: Bronstein, M., Weispfenning, V., Grabmeier, J. (eds) Symbolic Rewriting Techniques. Progress in Computer Science and Applied Logic, vol 15. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8800-4_11
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DOI: https://doi.org/10.1007/978-3-0348-8800-4_11
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