  In this paper we will define analogs of Gr\"obner bases for $R$-subalgebras
and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ 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 Gr\"obner basis counterparts. As
an application, we will produce a method for computing generators for the first
syzygy module of a subset of an $R$-subalgebra of $R[x_1,\ldots,x_n]$ where
each coordinate of each syzygy must be an element of the subalgebra.
