Ideals of Algebraic Sets

**slevvio** · October 9th 2011, 07:01 AM

Hello, I was wondering if someone could give me a hint about where to start with the following problem. Any help would be appreciated.

(definitions)
Let $k$ be an algebraically closed field. Then let $J,J' \subseteq k[x_1,\ldots,x_n]$ be ideals. Then let $X = V(J), X' = V(J')$ , i.e. algebraic sets. The ideal of $X$ is $I_X := \{ f \in k[x_1 ,\ldots, x_n] | f(x) = 0 \forall x \in X\}$ .

(problem)
give an example to show that $I_{X \cap X'} \not= I_X + I_{X'}$

**NonCommAlg** · October 9th 2011, 11:46 AM

Originally Posted by slevvio

Hello, I was wondering if someone could give me a hint about where to start with the following problem. Any help would be appreciated.

(definitions)
Let $k$ be an algebraically closed field. Then let $J,J' \subseteq k[x_1,\ldots,x_n]$ be ideals. Then let $X = V(J), X' = V(J')$ , i.e. algebraic sets. The ideal of $X$ is $I_X := \{ f \in k[x_1 ,\ldots, x_n] | f(x) = 0 \forall x \in X\}$ .

(problem)
give an example to show that $I_{X \cap X'} \not= I_X + I_{X'}$

this is just the geometric version of this question, that i've already answered.

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