Definition of "invariant in a module"

Question

What does it mean if someone say that the class of an ideal $I$ in a ring $R$ is an invariant of a module $M$?

Answer 1

In the reference you gave, it expresses $M$ as $$M\cong\sum(\overline{\theta}-\theta)^{e_i}U_iR$$ where the $U_i$ are ideals of $R_0$. It then says "the class of $\prod U_i$ is an invariant of $M$."

My best guess is that this means that if there is another representation

$$\sum(\overline{\theta}-\theta)^{e_i}V_iR$$ for ideals $V_i$ in $R_0$, then $\prod V_i$ and $\prod U_i$ have the same class.

I can't be 100% sure since it's possible they are using "invariant" in some specialized sense specific to this subtopic. There is certainly a lot of notation to catch up to at that point in the paper :)