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what does it mean, if you say, that the class of $\prod U_i$ in $R$ is an invariant of a module, where the $U_i$ are ideals in a ring $R$. I cannot find a definition. Thanks for help

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this is mentioned e.j. in Lemma 1.4 on page 217:… – james B. Jun 14 '12 at 11:56
Refer to the topic in the title please. – NikolajK Jun 14 '12 at 12:30

Something being an invariant of a module $M$ usually (and from a quick look also in this article) means that this object does not change if you take a different module $N$ isomorphic to $M$.

This article concerns a category where the Krull-Remak-Schmidt property does not hold, i.e. it could be that $M\cong \sum_{i=1}^n (\overline{\theta}-\theta))^{s_i}U_iR\cong \sum_{i=1}^n (\overline{\theta}-\theta))^{s_i}V_iR$, but $U_i\ncong V_i$. However, the lemma shows that $\prod U_i\cong \prod V_i$.

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