Let be a ring and an -module.

is the set of -homomorphisms from to .

Define by . Show is a ring isomorphism.

I know how to show it is a homomorphism and one-to-one, but having trouble with onto. Can I get some help?

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- Apr 4th 2010, 04:08 PMdori1123show ring isomorphism
Let be a ring and an -module.

is the set of -homomorphisms from to .

Define by . Show is a ring isomorphism.

I know how to show it is a homomorphism and one-to-one, but having trouble with onto. Can I get some help? - Apr 4th 2010, 06:39 PMaliceinwonderland
Your M seems to be a unitary R-module from your question because f(1) is defined.

I think it is difficult to deduce that is onto by using a alone.

Rather, define a map given by , where .

Suppose ( . Then, , because . Thus is well-defined. I'll leave it to you to show that is an R-module homomorphism.

Then and (verify this). - Apr 4th 2010, 08:24 PMFancyMouse
Just to show that given any element x in M, you can construct a well-defined R-homomorphism f from R to M such that f(1)=x. The function that aliceinwonderland constructed is exactly demonstrating this idea.