Let $h^*$ be a (multiplicative) generalized cohomology theory. Let $X$ be a CW complex which is a union of an increasing sequence $X_0 \subset X_1 \subset X_2 \subset \cdots$ of subcomplexes. Then there is a Milnor exact sequence $$0 \to (\varprojlim)^{1}(h^{n-1}(X_i)) \to h^n(X) \to \varprojlim h^n(X_i) \to 0.$$

Obviously, the restriction maps are ring maps and so the kernel $I=(\varprojlim)^{1}(h^{n-1}(X_i))$ gives an ideal.

Question: Is there anything that can be said about the multiplicative structure on the $(\varprojlim)^{1}(h^{n-1}(X_i))$ piece? In every example I find in textbooks, $I^2=0$, but I don't know if that's what happens in general.