the isomorphisms are defined very naturally: is defined by and is defined by:

for example to prove that is an R-module isomorphism: let and be in first see

that if and only if that is this shows that is well-defined and also it's one-to-one. next we show that is a homomorphism:

we also have: so is an R-module homomorphism. finally if then

let then obvioulsy thus is onto and we're done. an identical proof will work for