The Ideal of a Set of Points Review
The Ideal of a Set of Points Review
We will now review some of the recent material regarding the ideal of a set of points.
- On the [[[The Ideal of a Set of Points]] we said that if $K$ is a field and $X \subseteq \mathbb{A}^n(K)$ then the Ideal of $X$ is defined as the ideal:
\begin{align} \quad I(X) = \{ F \in K[x_1, x_2, ..., x_n] : F(\mathbf{p}) = 0, \: \forall \mathbf{p} \in X \} \end{align}
- That is, the ideal of $X$ is the ideal which contains all polynomials in $K[x_1, x_2, ..., x_n]$ that vanish at every point in $X$. We then verified that $I(X)$ is indeed an ideal in $K[x_1, x_2, ..., x_n]$.
- On the Basic Properties of the Ideal of a Set of Points page we looked at some properties of the ideal of a set of points. These properties are summarized below.
| Number | Property |
|---|---|
| 1 | If $X, Y \subseteq \mathbb{A}^n(K)$ and $X \subseteq Y$ then $I(X) \supseteq I(Y)$. |
| 2 | $I(\emptyset) = K[x_1, x_2, ..., x_n]$. |
| 3 | If $K$ is an infinite field then $I(\mathbb{A}^n(K)) = (0)$. |
- We then looked at some properties which relate $V$ and $I$:
| Number | Property |
|---|---|
| 4 | If $S \subseteq K[x_1, x_2, ..., x_n]$ then $S \subseteq I(V(S))$. |
| 5 | If $X \subseteq \mathbb{A}^n(K)$ then $X \subseteq V(I(X))$. |
- On the The Ideal of a Set of Points is a Radical Ideal page we defined an important type of ideal. We said that if $R$ is a ring and $I$ is an ideal then the Radical of $I$ is defined as:
\begin{align} \quad \mathrm{Rad}(I) = \{ a \in R : a^n \in I, \: \mathrm{for \: some \:} n \in \mathbb{N} \} \end{align}
- We proved that $\mathrm{Rad}(I)$ is indeed an ideal and said that an ideal $I$ is a Radical Ideal if:
\begin{align} \quad I = \mathrm{Rad}(I) \end{align}
- We then proved a nice result. We said that if $X \subseteq \mathbb{A}^n(K)$ then the ideal of $X$ is a radical ideal, that is:
\begin{align} \quad I(X) = \mathrm{Rad} (I(X)) \end{align}