I'm working on Exercise 2.6 from Atiyah-Macdonald's "Introduction to Commutative Algebra" and I had a question about the definition of bilinear maps.
Exercise 2.6: For any $A$-module $M$, let $M[x]$ denote the set of all polynomials in $x$ with coefficients in $M$, that is to say expressions of the form $$ m_0 + m_1x + \dots + m_r x^r \qquad (m_i\in M)$$ Defining the product of an element of $A[x]$ and an element of $M[x]$ in the obvious way, show that $M[x]$ is an $A[x]$-module. Show that $M[x] \cong A[x]\otimes_A M$ (as $A[x]$-modules).
So I begin by trying to find explicit $A[x]$-linear maps from $M[x]\to A[x]\otimes_A M$ and then from $A[x]\otimes_A M \to M[x]$. I got a little confused about the definition of a bilinear map when trying to define the second map.
What is an $A[x]$-bilinear map on $A[x]\otimes_A M$? In particular, let $\phi: A[x]\times M \to M[x]$ be a map. Suppose I fix a first coordinate $f\in A[x]$, then what does it mean for $\phi(f,-) : M\to M[x]$ to be $A[x]$-linear? A priori, there is no defined multiplication of $A[x]$ onto $M$ right?