Polynomial ring with uncountable indeterminates In Rotman's Advanced Modern Algebra second edition (2010), on page 883 (or on page 905 in its first edition (2002)), in the proof of the existence of localization of a commutative ring $R$ on its multiplicative subset $S$, he writes:  
"Let $X=(x_{s})_{s\in S}$ be an indexed set with $x_{s}\mapsto s$ a bijection $X \rightarrow S$, and let $R[X]$ be the polynomial ring over R with indeterminates $X$."
However in his definition of formal power series over $R$ he comments:  
"To determine when two formal power series are equal, let us recognize that a sequence $\sigma$ is really a function $\sigma:\mathbb{N} \rightarrow R$, where $\mathbb{N}$ is the set of  natural numbers, with $\sigma(i) = s_{i}$ for all $i \geq 0$."
So I want to ask if $S$ is uncountable, then is it still legitimate that he defines $R[X]$ in this way?
 A: If $X$ is an arbitrary set, then $R[X]$ is defined to be the free commutative $R$-algebra over $X$. It is also known as the polynomial ring (better would be polynomial algebra) with variables $X$ and coefficients $R$. You can construct it as the symmetric algebra over the free module over $X$, or equivalently as the monoid algebra over the free commutative monoid over $X$. Explicitly, it consists of maps $\sigma : \mathbb{N}^{(X)} \to R$ with finite support, where we think of this as the polynomial $\sum_{\alpha \in \mathbb{N}^{(X)}} \sigma(\alpha) \cdot X^{\alpha}$. The formal power series ring $R[[X]]$ is the $(X)$-adic completion of $R[X]$. Its elements are maps $\sigma : \mathbb{N}^{(X)} \to R$ such that for every $d \in \mathbb{N}$ the set $\{\alpha \in \mathbb{N}^{(X)} : \sum_x \alpha_x = d,~ \sigma(\alpha) \neq 0\}$ (monomials of degree $d$ appearing in the power series) is finite. Of course, nothing has to be assumed countable here, and your quote refers to the single variable case $X=\{x\}$.
