Gröbner bases for modules over Prüfer domains

Abstract

Let R be a Prüfer domain of Krull dimension one. We prove the existence of Gröbner 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 Gröbner 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 Gröbnerr basis is actually a strong Gröbner basis.