I am thinking the generated subring of polynomial ring $K[X]$ where $K$ is a field. First I find that it is isomorphism to a quotient ring of $Z[Y]$. For example, let $S = \{f_1,f_2\}$ and $f_1, f_2\in K[X]$，and $I=\{g\in Z[y_1,y_2]|g(f_1,f_2)=0\}$. It can verify that $I$ is an ideal of $Z[y_1,y_2]$ and the subring generated by $S$ is isomorphism to $Z[y_1,y_2]/I$. Since $Z[y_1,y_2]$ is a Noetherian ring, $I$ is generated by finite elements of $Z[y_1,y_2]$ and there is a Gröbner basis $G$ of $I$.
My question is: how to compute the Gröbner basis $G$ for a given $S$? Is there some method or research about this?