I have some questions about basic facts of invariant ideals. First the definition: given an Hopf algebroid $(A,\Gamma)$ an ideal $I$ of $A$ is called invariant if $\eta_R(I)=\eta_L(I)$ (see Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Appendix A1).
I want to understand what is the concept behind the definition: given such an ideal $I$ do we have that $A/I$ or $I$ inherit a structure of left comodule from the usual $\eta_R \colon A \rightarrow \Gamma$? I tried to write down the square diagram associated to this condition but I do not see why the equation $\eta_R(I)=\eta_L(I)$ is required for the commutativity.
Then concrete examples are given by $(X_*, X_*(X))$ for a spectrum $X$ (e.g. $MU$, $BP$ or $H\mathbb{F}_p$): in this situation we have that $X^*(X)=[X,X]_*$ has a natural action on $X_* = [ \mathbb{S}, X]_*$, is the previous definition equivalent to requiring that $I$ is invariant with respect this action?
Thanks in advance.