Intersection of ideals in a polynomial ring over a dual valuation domain
A dual valuation ring is a ring of the form \(\mathbf{A}= \mathbf{V} + \epsilon \mathbf{V} \cong \mathbf{V}[\epsilon]/\langle{\epsilon^2}\rangle\), where \(\mathbf{V} \) is a valuation domain. The authors propose a method for computing strong Gröbner bases for finitely-generated ideals of \(\mathbf{A}[X_1,\ldots,X_n]\). As application, they compute finite generating sets for syzygy modules over \(\mathbf{A}[X_1,\ldots,X_n]\) as well as for the intersection of two finitely-generated ideals of \(\mathbf{A}[X_1,\ldots,X_n]\). Since it is now known that the existence of finite Groebner bases for finitely-generated ideals of \(\mathbf{V}[X_1,\ldots,X_n]\) (\(\mathbf{V}\) being a valuation domain) is only guaranteed in case \(\mathbf{V}\) has Krull dimension \(\leq 1\), the authors may have forgotten to mention that condition.
scientific article; zbMATH DE number 7029808