Abstract

Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the term order is POT, or, “position over term”. In order to do this, we first prove that there is a Gr¨obner basis for finitely generated ideals in R[X], which is a special case of the main result. The proof is based on the results from [3]. In addition to this we show, in the case of valuation domains, that every Gr¨obner basis is actually a strong Gr¨obner basis

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