Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

Abstract

This work's primary objective is to investigate an effective Buchberger-Zacharias-Weispfenning theory of bimodules over skew polynomial extensions of a coefficient ring satisfying the appropriate notion of coherence. The original contribution of our approach establishes the subbilateral coherence together with the effective computation of subbilateral Gröbner bases for a sufficiently large class of skew polynomial extensions, over term-ordered monoids, of subbilateral Noetherian coherent coefficient rings. The strategy to produce finite bilateral Gröbner bases from subbilateral ones generalizes Kandri-Rody and Weispfenning completion from left Gröbner bases to bilateral ones over a subclass of solvable polynomial rings, as well as Weispfenning completion from Weispfenning restricted modules to bilateral ones over the bivariate Weispfenning extensions of skew fields. It equally results that the class of rings that are both left Noetherian and left coherent (or resp., right Noetherian and right coherent) is closed under iterative skew polynomial extension with bijective conjugation maps.