Direct-sum cancellation for modules over real quadratic orders

Let R be an order in a real quadratic number field. We say that R has mixed cancellation, respectively, torsion-free cancellation if L⊕M≅L⊕N⇒M≅N holds for all finitely generated R-modules M, N and L, respectively, for all finitely generated torsion-free R-modules M, N and L. We derive criteria for real quadratic orders to have mixed cancellation. For instance, we prove that torsion-free cancellation holds and mixed cancellation fails for all orders Rp≔Z[p1+p2], where p is a prime satisfying 13≤p≤1011 and p≡1mod4. Our considerations show that if the Ankeny-Artin-Chowla conjecture turned out to be true, then Rp would have torsion-free cancellation but not mixed cancellation for every prime p≥13 with p≡1mod4.