$\begingroup$

Let $R$ be a ring, $I$ an ideal, and $\langle g_1, \ldots, g_m \rangle$ a finitely generated ideal. Considering the intersection $I \cap \langle g_1, \ldots, g_m \rangle$, I became interested in the set of coefficients $c \in R^m$ such that $\sum_i c_i g_i \in I$.

Question: is there an algorithm to calculate $\{ c \in R^m : \sum_i c_i g_i \in I \}$?

It seems to be an $R$-module. It is easy to see that $\prod_i (I : (g_i))$ is contained in it (which contains $IR^m$).

Does it have a name? Can it be expressed in terms of some other objects which can be computed?

I am interested in the case $R = k[x_1,\ldots,x_n]$ where $k$ is a field, and (so) the ideal $I$ is also finitely generated. Maybe a Groebner basis of the intersection could help.

Solution attempt: Suppose $I \cap \langle g_1, \ldots, g_m \rangle$ is finitely generated (e.g. $R$ is Noetherian). Say it is equal to $\langle h_1, \ldots, h_t \rangle$, and express $h_s = \sum_i b_{s,i} g_i$ with $b_{s,i} \in R$. Then $b_s = (b_{s,i}) \in R^m$ is contained in the set I'm interested in, for each $s=1,\ldots,t$. So the span $Rb_1+ \ldots+ Rb_t \subset R^m$ is contained in the set I'm interested in.

Now the question is whether a different choice of basis for the intersection (or a different choice of coefficients) can give more elements in the span, and whether $Rb_1 + \ldots + Rb_t$ exhausts the set I'm interested in.