I had a similar train of thought as these two questions, Operations on Ideals in Terms of Generators , LCM generators for the intersection of non-principal ideals in a Noetherian UFD , trying to think of what the generators of an intersection of ideals could look like in terms of the original generators.
I was considering the following:
Let $I_1$ and $I_2$ be two finitely generated ideals of an arbitrary commutative ring $R$. And let $A_1 = \{ a_1,\dots, a_s \}$ and $A_2=\{ b_1,\dots, b_t\}$ be the corresponding finite sets of generators.
Then let $S_1=\{ a | a \in A_1, a \in I_2 \}$ be the set of generators of $I_1$ that are also in $I_2$. And similarly let $S_2=\{ b | b \in A_2, b \in I_1 \}$.
Define the ideal $J$ as follows:
Handle some edge cases:
(Or maybe these could be absorbed into the general definition below with a little bit of rearranging, or just defining how to handle lcm products with an empty set)if $A_1 \setminus S_1$ is empty, then let $J = I_1$.
if $A_2 \setminus S_2$ is empty, then let $J = I_2$.
$J$ is the ideal generated by
$$S_1 \cup S_2 \cup \{ \text{lcm}(a, b) | a \in A_1 \setminus S_1,\ b \in A_2 \setminus S_2 \} .$$
I realize this construction doesn't make calculating $J$ simple if we only have the generators, as we may need to get the Groebner basis to determine inclusions like $a \in I_2$, etc. But in some cases with additional restrictions (eg. ideal of a polynomial ring with homogeneous basis) those are easier to calculate than a full Groebner basis. Anyway, that is a digression for the question at hand.
By construction of $J$, everything in $J$ is also in $I_1$ and $I_2$. As the product can be written as the ideal generated by $\{ a b ∣ a \in A_1, b \in A_2 \}$, then everything in the product $I_1 I_2$ is in $J$. So it appears we have:
$$ I_1 I_2 \subseteq J \subseteq I_1 \cap I_2 $$
When $I_1 = I_2$, $J$ is the intersection, not the product. So I was hoping I could prove that $J$ equals the intersection. Then I found the answers to these related questions Generators for the intersection of two ideals , Intersection of finitely generated ideals , which show that the intersection of finitely generated ideals may not necessarily be finitely generated itself.
So there must be cases where $J$ is not the intersection of $I_1$ and $I_2$.
So then what is this finitely generated $J$ ?
Besides some edge cases handling lcm products with an empty set, this didn't feel that contrived to me. But maybe that is enough to make it contrived "junk" with no simple interpretation.
Or, my hope, is that in some sufficiently "nice" cases (in particular I hope for polynomials with finite number of variables over some field), that $J$ is indeed the intersection?
Understanding where this can fall between the product and the intersection in the general case would be nice, but I'd really love any explanation or counter-example which clarifies if it is the intersection in the nice polynomials over a field case.
