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

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- November 13th 2008, 02:07 PMErdos32212[SOLVED] R-module isomorphisms, R[x]
Let R be a ring. \prod_{i=0}^{\infty} R[tex].

- November 13th 2008, 06:41 PMNonCommAlg
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