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

  • Thread starter RVP91
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In summary, the problem is to prove that HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K)). This is done by defining a map, f, from HomA(M,HomA(N,K)) to HomA(N,HomA(M,K)) and showing that it is an isomorphism of A-modules. The next step is to define another map in the opposite direction and show that it is the inverse of f. The objects N, M, and K are A-modules.
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RVP91
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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!
 
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  • #2
What are N,M,K?
 
  • #3
A-modules.
 

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

1. What does the statement "HomA(M,HomA(N,K)) is isomorphic to HomA(N,HomA(M,K))" mean?

The statement means that there is a bijective function between the two sets of homomorphisms, meaning that they have the same number of elements and the same structure.

2. Can you provide an example to explain this statement?

Sure, let's say we have two groups M and N, and a ring K. HomA(M,HomA(N,K)) would be the set of all homomorphisms from M to HomA(N,K), which is the set of all homomorphisms from N to K. Similarly, HomA(N,HomA(M,K)) would be the set of all homomorphisms from N to HomA(M,K), which is the set of all homomorphisms from M to K. In this case, we can see that the two sets are isomorphic, as there is a bijective function between them.

3. Why is this statement important in mathematics?

This statement is important because it shows that the order in which we apply homomorphisms does not matter, as long as the underlying structures are the same. This can be useful in simplifying calculations and proofs in various areas of mathematics.

4. Is this statement true for all types of algebraic structures?

No, this statement is not true for all types of algebraic structures. It is specifically true for groups and rings, but may not hold for other structures such as fields or vector spaces.

5. How can this statement be proved?

This statement can be proved using the definition of homomorphisms and the properties of isomorphisms. By showing that there is a bijective function between the two sets of homomorphisms, we can prove that they are isomorphic.

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