# algebraic sets and polynomial ideals

Suppose $k$ is a field. Let $\mathbb{A}^{n}_{k}$ denote affine $n$-space over $k$.
For $S\subseteq k[x_{1},\ldots,x_{n}]$, define $V(S)$, the zero set of $S$, by

 $V(S)=\{(a_{1},\ldots,a_{n})\in k^{n}\mid f(a_{1},\ldots,a_{n})=0\text{ for all% }f\in S\}$

We say that $Y\subseteq\mathbb{A}^{n}_{k}$ is an (affine) algebraic set if there exists $T\subseteq k[x_{1},\ldots,x_{n}]$ such that $Y=V(T)$. Taking these subsets of $\mathbb{A}^{n}_{k}$ as a definition of the closed sets of a topology induces the Zariski topology over $\mathbb{A}^{n}_{k}$.
For $Y\subseteq\mathbb{A}^{n}_{k}$, define the deal of $Y$ in $k[x_{1},\ldots,x_{n}]$ by

 $I(Y)=\{f\in k[x_{1},\ldots,x_{n}]\mid f(P)=0\text{ for all }P\in Y\}.$

It is easily shown that $I(Y)$ is an ideal of $k[x_{1},\ldots,x_{n}]$.
Thus we have defined a function $V$ mapping from subsets of $k[x_{1},\ldots,x_{n}]$ to algebraic sets in $\mathbb{A}^{n}_{k}$, and a function $I$ mapping from subsets of $\mathbb{A}^{n}$ to ideals of $k[x_{1},\ldots,x_{n}]$.
We remark that the theory of algebraic sets presented herein is most cleanly stated over an algebraically closed field. For example, over such a field, the above have the following properties:

1. 1.

$S_{1}\subseteq S_{2}\subseteq k[x_{1},\ldots,x_{n}]$ implies $V(S_{1})\supseteq V(S_{2})$.

2. 2.

$Y_{1}\subseteq Y_{2}\subseteq\mathbb{A}_{k}^{n}$ implies $I(Y_{1})\supseteq I(Y_{2})$.

3. 3.

For any ideal $\mathfrak{a}\subset k[x_{1},\ldots,x_{n}]$, $I(V(\mathfrak{a}))=\operatorname{Rad}(\mathfrak{a})$.

4. 4.

For any $Y\subset\mathbb{A}^{n}_{k}$, $V(I(Y))=\overline{Y}$, the closure of $Y$ in the Zariski topology.

From the above, we see that there is a 1-1 correspondence between algebraic sets in $\mathbb{A}^{n}_{k}$ and radical ideals of $k[x_{1},\ldots,x_{n}]$. Furthermore, an algebraic set $Y\subseteq\mathbb{A}^{n}_{k}$ is an affine variety if and only if $I(Y)$ is a prime ideal. As an example of how things can go wrong, the radical ideals $(1)$ and $(x^{2}+1)$ in $\mathbb{R}[x]$ define the same zero locus (the empty set) inside of $\mathbb{R}$, but are not the same ideal, and hence there is no such 1-1 correspondence.

 Title algebraic sets and polynomial ideals Canonical name AlgebraicSetsAndPolynomialIdeals Date of creation 2013-03-22 13:05:40 Last modified on 2013-03-22 13:05:40 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 16 Author mathcam (2727) Entry type Definition Classification msc 14A10 Synonym vanishing set Related topic Ideal Related topic HilbertsNullstellensatz Related topic RadicalOfAnIdeal Defines zero set Defines algebraic set Defines ideal of an algebraic set Defines affine algebraic set