Gröbner bases for ideals in univariate polynomial rings over valuation rings

Abstract

Let V be a valuation ring such that dim(V ) = 0 and the annihilator of each element in V is finitely generated. In this paper it is proved that if I is a finitely generated ideal in the polynomial ring V [X], then there is a Gröbner basis for I. Also, an example of a zero-dimensional non-Noetherian valuation ring RM is presented, together with an example of finding a Gröbner basis for a certain ideal in a polynomial ring RM [X].