Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 3 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.

You may also find this link and this link helpful.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.