Anti-Isomorphisms and the Failure of Duality

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