In this talk we present the Buchberger’s algorithm for computing Gröbner bases of modules defined on a new class of noncommutative rings: the skew $PBW$ extensions, introduced by us in , as a generalization of the $PBW$ extensions established by Bell and Goodearl in . Further, we show some elementary applications of this, such as: membership problem, syzygy module, presentation of a module, kernel and image of a homomorphism.
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