Analogs of Gröbner Bases in Polynomial Rings over a Ring

In this paper we will define analogs of Grobner bases for R -subalgebras and their ideals in a polynomial ring R [ x 1 ,..., x n ] where R is 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 Grobner basis counterparts.As an application, we will produce a method for computing generators for the first syzygy module of a subset of an R [ x 1 ,..., x n ] where each coordinate of each syzygy must be an element of the subalgebra.