HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K))

[RVP91]

Homework Statement

Let A be a commutative ring with identity element.

Prove that HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K)).

Homework Equations

The Attempt at a Solution

I believe it is best to start by defining a map, f: HomA(M,HomA(N,K) → HomA(N,HomA(M,K))
for ψ: M → HomA(N,K) so that f(ψ)(n): m → ψ(m)(n).

Then I guess I need to show this is an isomorphism of A-modules.

However I'm not sure how to proceed. In similar questions I have defined another map usually in the opposite direction and shown they are inverse. This time though I'm not sure where to go.


Any help would be appreciated.

Thanks in advance!

 

[MathematicalPhysicist]

What are N,M,K?

 

[RVP91]

A-modules.