On page 658 of Dummit & Foote, they write:

Recall that the set $\mathbb A^n$ of $n$-tuples of elements of the field $k$ is called affine $n$-space over $k$. If $x_1,x_2,\dots,x_n$ are independent variables over $k$, then the polynomials $f$ in $k[x_1,x_2,\dots,x_n]$ can be viewed as $k$-valued functions $f\colon \mathbb A^n \to k$ on $\mathbb A^n$ by evaluating $f$ at the points in $\mathbb A^n$: $$ f\colon (a_1,a_2,\dots,a_n) \mapsto f(a_1,a_2,\dots,a_n)\in k. $$ This gives a ring of $k$-valued functions on $\mathbb A^n$, denoted by $k[\mathbb A^n]$ and called the coordinate ring of $\mathbb A^n$.

They go on to use $k[\mathbb A^n]$ and $k[x_1,x_2,\dots,x_n]$ interchangeably. However, I think that these two rings are not necessarily isomorphic. For example, if $k = \mathbb Z/2\mathbb Z$, then while the polynomials $x$ and $x^2$ are distinct elements of $k[x]$, they correspond to the same function from $\mathbb A^1 \to k$, namely, the identity. So, how should I be thinking about $k[\mathbb A^n]$? In particular, in this example should $x$ and $x^2$ be thought of as distinct or the same elements of $k[\mathbb A^1]$?

First update: Later on in the same section they give the following definitions:

In general, for any subset $A$ of $\mathbb A^n$ define $$ \mathcal I(A) = \{ f\in k[x_1,\dots,x_n] \mid f(a_1,a_2,\dots,a_n) = 0 \text{ for all } (a_1,a_2,\dots,a_n) \in A \}. $$


If $V\subseteq \mathbb A^n$ is an affine algebraic set the quotient ring $k[\mathbb A^n]/\mathcal I(V)$ is called the coordinate ring of $V$, and is denoted by $k[V]$.

However, Wikipedia and Wolfram define the coordinate ring of $V\subseteq \mathbb A^n$ as being simply $$ \frac{k[x_1,\dots,x_n]}{\mathcal I(V)}, $$ where $\mathcal I(V)$ denotes the ideal of all polynomials in $k[x_1,x_2,\dots,x_n]$ that vanish at all points of $V$.

I feel like Dummit and Foote are trying to arrive at the same place, but it's quite confusing when they use coordinate ring in the definition of a coordinate ring.

Second update: I think the following quotation from an MIT lecture best clarifies what's going on:

Recall that $k[V]$ is defined as the collection of polynomial functions $\phi\colon V\to k$.

We will identify $k[V]$ with the quotient ring $k[x_1,\dots,x_n]/ \mathcal I(V)$, using the isomorphism $$ k[V] \cong \frac{k[x_1,\dots,x_n]}{\mathcal I(V)}. $$

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    $\begingroup$ Either $k$ should be infinite or you should use a more refined definition of affine space. The answer to aim for is the polynomial ring. $\endgroup$ – Qiaochu Yuan Mar 24 '16 at 21:06
  • $\begingroup$ @QiaochuYuan Thanks. I edited to include two more excerpts to make Dummit and Foote's presentation clearer... the definitions on Wikipedia and Wolfram (which I also refer to in the edit) make much more sense to me since they don't feel as circular. $\endgroup$ – justin Mar 24 '16 at 21:16

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