Analogs of Gröbner Bases in Polynomial Rings over a Ring

AbstractIn this paper we will define analogs of Gröbner bases forR-subalgebras and their ideals in a polynomial ringR[x1,...,xn] whereRis a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies.The main goal is to present and verify algorithms for constructing these Gröbner basis counterparts.As an application, we will produce a method for computing generators for the first syzygy module of a subset of anR[x1,...,xn] where each coordinate of each syzygy must be an element of the subalgebra