algebraic sets and polynomial ideals

Suppose $k$ is a field. Let ${𝔸}_k^n$ denote affine $n$-space over $k$.
For $S\subseteq k[x_1,\mathrm{\dots },x_n]$, define $V(S)$, the zero set of $S$, by

$$V(S)=\{(a_1,\mathrm{\dots },a_n)\in k^n\mid f(a_1,\mathrm{\dots },a_n)=0\text{ for all }f\in S\}$$

We say that $Y\subseteq {𝔸}_k^n$ is an (affine) algebraic set if there exists $T\subseteq k[x_1,\mathrm{\dots },x_n]$ such that $Y=V(T)$. Taking these subsets of ${𝔸}_k^n$ as a definition of the closed sets of a topology induces the Zariski topology over ${𝔸}_k^n$.
For $Y\subseteq {𝔸}_k^n$, define the deal of $Y$ in $k\mathit{}\mathrm{[}{x}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_n\mathrm{]}$ by

$$I(Y)=\{f\in k[x_1,\mathrm{\dots },x_n]\mid f(P)=0\text{ for all }P\in Y\}.$$

It is easily shown that $I(Y)$ is an ideal of $k[x_1,\mathrm{\dots },x_n]$.
Thus we have defined a function $V$ mapping from subsets of $k[x_1,\mathrm{\dots },x_n]$ to algebraic sets in ${𝔸}_k^n$, and a function $I$ mapping from subsets of ${𝔸}^n$ to ideals of $k[x_1,\mathrm{\dots },x_n]$.
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. $S_1\subseteq S_2\subseteq k[x_1,\mathrm{\dots },x_n]$ implies$V(S_1)\supseteq V(S_2)$.

2. $Y_1\subseteq Y_2\subseteq {𝔸}_k^n$ implies$I(Y_1)\supseteq I(Y_2)$.

3. For any ideal $𝔞\subset k[x_1,\mathrm{\dots },x_n]$,$I(V(𝔞))=\mathrm{Rad}(𝔞)$.

4. For any $Y\subset {𝔸}_k^n$, $V(I(Y))=\overline{Y}$, the closureof $Y$ in the Zariski topology.

From the above, we see that there is a 1-1 correspondence between algebraic sets in ${𝔸}_k^n$ and radical ideals of $k[x_1,\mathrm{\dots },x_n]$. Furthermore, an algebraic set $Y\subseteq {𝔸}_k^n$ 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 $ℝ[x]$ define the same zero locus (the empty set) inside of $ℝ$, 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