Let us consider $K[x_1,\ldots,x_n]$, the ring of multivariate polynomials in variables $x_1,\ldots,x_n$ with coefficients in some field $K$. With this, we consider the subring $R$ generated by a finite set $G \subseteq K[X_1,\ldots,X_n]$. That is, $R$ is the set of polynomials which can be expressed as polynomials in $G$. For instance, if $G = \{ g_1, g_2 \} = \{ x_1^2 + x_2^2, x_3 \}$, then $(x_1^2 + x_2^2) \cdot x_3 - x_3^2 \in R$ because $g_1^2 g_2 - g_2^2 \in R$; instead, $x_1^2 \notin R$.
In the case of ideals, there are Gröbner bases and Buchberger's algorithm. I was wondering whether there are algorithmic counterparts for subrings. Specifically, given an arbitrary $p \in K[x_1,\ldots,x_n]$, are there any established algorithms which would compute some "multivariate remainder $r$ of $p$ modulo $G$"? For instance, some (non-unique) $r$ such that $p = f(g_1,\ldots,g_n) + r$, where $f$ is a polynomial, $g_1,\ldots,g_n \in G$ and $r$ has minimal degree.
Any hint or reference would be appreciated.