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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.
Goldsmith, Brendan; Corner, A. L. S.; and Wallutis, S., "Anti-isomorphisms and the failure of duality" (2008). Book chapter/book. Paper 2.