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Let $k$ be an algebraic closed field and $k[x,y,z]$ be the polynomial ring.

Now consider the ideal $I=(xy,yz,zx)$ in $k[x,y,z]$. My question is, can the ideal $I$ be generated by two elements?

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up vote 2 down vote accepted

No, it is not possible. The key idea is to use that the ring is graded and the ideal is homogeneous.

Let $S=k[x,y,z],$ suppose otherwise, and let $(f,g)=I.$ We have $I_2=I\cap S_2=\operatorname{Span}_k\langle xy,yz,zx\rangle$ from the first set of generators.

On the other hand, letting $f_i,g_i$ denote the degree $i$ parts of $f,g,$ we know that $f_i,g_i\in I,$ since $I$ is homogeneous. This implies that we must have $f_0,f_1,g_0,g_1$ all equal to zero, since we know that $\dim_k(I_0)=\dim_k(I_1)=0$ from the first set of generators.

Since $f_2,g_2\in I_2$ and $f,g$ generate $I,$ this shows that $\dim_k(I_2)=2,$ since we only "stay inside" $I_2$ by multiplying by a degree zero element.

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