Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ such that $$I_a \subseteq \sum_{A \ni b\neq a} I_b.$$ A set is independent if it is not dependent. For example two ideals are dependent if and only if one is contained in the other. I would like to know if this notion of (in)dependence is meaningful and if there is literature about it.

For example, the ideals $(x), (y), (x+y)$ define the uniform matroid $U_{2,3}$ since any two of them are independent, but any of them is contained in the sum of the other two (which equals $(x,y)$). However, the resulting notion is not 'matroid' -- take any three ideals $I_1,I_2,I_3$ such that $I_1\subset I_2$, $I_1 \subset I_3$, but $I_2,I_3$ incomparable. Then $\{I_1\}$ and $\{I_2, I_3\}$ are maximal independent sets of different sizes.

This problem comes from an application in algebraic statistics. There each ideal comes from a conditional independence statement for certain random variables and a collection of statements corresponds to the sum of the ideals. One would like to say things like "a colletion of statements is (ir)redundant", but is there a well defined notion?