GRÖBNER BASES FOR IDEALS IN UNIVARIATE POLYNOMIAL RINGS OVER VALUATION RINGS

Abstract

Let $V$ be a valuation ring such that $\mathrm{dim}(V)=0$ 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 $R_M$ is presented, together with an example of finding a Gröbner basis for a certain ideal in a polynomial ring $R_M[X]$.