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Groups and modules with isomorphic endomorphism rings are known, in certain cases, to be necessarily isomorphic. When such a ring isomorphism is replaced by an anti-isomorphism, the modules are often determined only up to isomorphism of certain duals. This type of situation is examined in a number of cases with special emphasis on the situation for mixed Abelian groups, where it is shown that no reasonable duality may exist.
Corner, A.L.S., Goldsmith, B. & Wallutis, S.L. (2008). Anti-Isomorphisms and the Failure of Duality. Models, Modules and Abelian Groups , DeGruyter. https//doi.org/10.1515/9783110203035.315