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?
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).