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The result you mention is a classical result on Hopf modules, first proved by Larson and Sweedler. My favorite reference is

Pareigis: When Hopf Algebras are Frobenius Algebras. J. of Alg. 18(1971), 588-596. Lemma 2.

There you can also find the maps you describe in your question (so the answer is yes, they are inverse to each other).

Another reference is Sweedler's book (mentioned already by Mariano), Theorem 4.1.1. However, I think in the book the base ring is always a field, while Pareigis works over a comm. ring.

Somewhere in the book "Brzezinski, Wisbauer: Corings and Comodules" I read that the result in question also follows from a more general theorem on comodules over corings. But I don't know details.

1

The result you mention is a classical result on Hopf modules, proved by Larson and Sweedler. My favorite reference is

Pareigis: When Hopf Algebras are Frobenius Algebras. J. of Alg. 18(1971), 588-596. Lemma 2.

There you can also find the maps you describe in your question (so the answer is yes, they are inverse to each other).

Another reference is Sweedler's book (mentioned already by Mariano), Theorem 4.1.1. However, I think in the book the base ring is always a field, while Pareigis works over a comm. ring.

Somewhere in the book "Brzezinski, Wisbauer: Corings and Comodules" I read that the result in question also follows from a more general theorem on comodules over corings. But I don't know details.