I'm studying Atiyah's commutative algebra. I have a question with free modules and the kind of thing in polynomial ring. I wrote the following so it cannot be true facts.
A free $A$-module is $\bigoplus_{i\in I} A$. Then we know that every module $M$ is a factor module of a free module since there is a $A$-module homomorphism $\phi : \bigoplus_{x \in M} A \to M$ by $ (a_x) \mapsto \sum a_x x$ so that $M \simeq \bigoplus A / ker(\phi)$.
And we know that a finitely generated $A$-algebra $B$ is a quotient of a polynomial ring, $B \simeq A[t_1,\cdots,t_n]/kernel$. Then for non finitely generated case, I guess there exists similar concept of a free module, i.e. $A[t_i|t_i \in B]$. What we call it? Is it just a polynomial ring on infinitely many generators? Or is it something named with "free"? (I'm not familiar with the free objects.)