Assume we have a free ring $S$ of rank 2 and $K= S \otimes Q$. Now if we define an equivalent relation similar to the case $S=O_{K}$ then is it true that the $S$-ideal classes form a group ?

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everyproper subring $S$ of the integers of a quadratic field, besides the subring ${\mathbf Z}$, is free of rank 2 as a ${\mathbf Z}$-module and contains a non-invertible ideal, so the $S$-ideal classes are not a group under multiplication. – KConrad Sep 26 '11 at 12:06