Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$. Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\ldots f_m$.

Question: Is there an algorithm which give a generating set for the intersection $B\cap I$? In other words, if we consider the presentation of the commutative algebra $A/I$ arising from $I$, can we also get a nice presentation for its subalgebra $B/(B\cap I)$ ?