Let R be a ring. \prod_{i=0}^{\infty} R[tex].
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