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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.)

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Yes, you are correct, it is a polynomial ring on infinitely many generators. Given any set $T=\{t_1,t_2,\ldots\}$, we can form the free $A$-algebra generated by $T$, denoted by $A[t_1,t_2,\ldots]$ or just $A[T]$, which is the polynomial ring with coefficients in $A$ and with the elements of $T$ acting as indeterminates. This construction does not depend on $T$ being finite or infinite. In fact we could take $T$ to be uncountable. An important thing to keep in mind is that while there could be infinitely many elements of $T$, any single element $f(t_i)\in A[T]$, i.e. a polynomial in the symbols $t_i$ with coefficients in $A$, will only use finitely many of the $t_i$.

When $B$ is any $A$-algebra, $B$ is a quotient of $A[\{t_b\}_{b\in B}]$ by sending, for example, $a_1t_{b}+a_2t_{3b}^2$ to $a_1b+a_2(3b)^2=a_1b+9a_2b^2\in B$.

For example, the ring $\mathbb{C}$ is a $\mathbb{Q}$-algebra. We map $\mathbb{Q}[\{t_{\alpha}\}_{\alpha\in\mathbb{C}}]$ to $\mathbb{C}$ by sending each $\mathbb{Q}$-polynomial $f(t_{\alpha},t_{\beta},\ldots)$ to its evaluation when we replace $t_\alpha$ with $\alpha\in\mathbb{C}$, $t_\beta$ with $\beta\in\mathbb{C}$, etc., so that $\mathbb{C}$ is isomorphic to the quotient of $\mathbb{Q}[\{t_{\alpha}\}_{\alpha\in\mathbb{C}}]$ by the kernel of the evaluation map.

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