Co-Frobenius coalgebras

We investigate left and right co-Frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of Frobenius algebras. We prove that a coalgebra is left and right coFrobenius if and only if C congruent to Rat(C*(C*)) as right C*-modules and also that this is equivalent to the fact that the functors Hom(K) (-, K) and Hom(C*) (-, C*) from M-C to M-C* are isomorphic. This allows a definition of a left-right symmetric concept of co-Frobenius coalgebras that is perfectly dual to the one of Frobenius algebras and coincides to the existing notion left and right co-Frobenius coalgebra. (c) 2006 Elsevier Inc. All rights reserved.