TITLE: Tensor products and polynomials with coefficients in a module
QUESTION [5 upvotes]: This is exercise $6$ in chapter $2$ on modules from Atiyah's and Macdonald's book.

Let $M$ be an $A$-module and let $M[x]$ be the set of all polynomials in $x$ with coefficients in $M$. Then $M[x]$ has structure of $A[x]$-module.
Show that $M[x] \cong A[x] \otimes_{A} M$.

Some questions:
1) We can view $M[x]$ as an $A$-module yes? via the map $A \times M[x] \rightarrow M[x]$ given by $(a,m_{0}+\cdots+m_{n}x^{n}) \mapsto am_{0}+\cdots+a_{n}m_{n}x^{n}$.
2) Can we proceed as follows? Note that $A[x] \cong \oplus A$ (infinite direct sum), note then:
$A[x] \otimes_{A} M \cong \oplus A \otimes_{A} M \cong \oplus (A \otimes_{A} M)$.
But $M$ is an $A$-module so $A \otimes_{A} M \cong M$.
Therefore $A[x] \otimes_{A} M \cong \oplus M$. But I think we also have that:
$M[x] \cong \oplus M$ as $A$-modules right?
So $A[x] \otimes_{A} M \cong M[x]$. 
Is this OK?

REPLY [3 votes]: Point 1): yes, you can do this.  It's a "change of rings" arising from the ring map $A \to A[x]$ sending $a \in A$ to the constant polynomial $a$.
Point 2): as others pointed out, if you want an isomorphism of $A[x]$-modules what you wrote is not enough.  If you would like to do this in a fancy way you can show that $M[x]$ satisfies the universal property of the induced module $M\uparrow _A ^{A[x]} = A[x] \otimes _A M$.  If you prefer to do it by down-to-earth methods, let $\phi: A[x]\otimes_A M \to M[x]$ be defined by extending $ax^n \otimes m \mapsto amx^n$.  You need to check this is a well-defined $A[x]$-module map and that it has an inverse described by $mx^n \mapsto x^n \otimes m$.