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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?

user2690457
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    Note that in the first quote, you only have a polynomial ring over $R$ not formal power series over $R$. There $X$ is just the set of your indeterminates. – Tigran Hakobyan Nov 18 '13 at 06:11
  • The problem is that he defines polynomial ring over $R$ to be a formal power series over $R$ with finitely nonzero coefficients. But formal power series has at most countable indeterminates, the case here is that $X$ could be uncountable, which does not directly fit into the definition. – user2690457 Nov 18 '13 at 06:23

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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 $X=\{x\}$.

Martin Brandenburg
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