Consider complex polynomial ring $R=\mathbb{C}[x_1,x_2,x_3,x_4]$. By Hilbert's basis theorem it's a Noetherian ring. So I am wondering whether there exists an ideal that has no finite basis in itself. My teacher told me that it is impossible since there are only finite variants (four variants: $x_1,x_2,x_3,x_4$). But he didn't show me why. And either could I prove his statement or find an counterexample.
Actually I tried to construct a counterexample like $\{x_{i}^{p}+x_{j}^p\mid 1\leq i<j\leq4,\ p\text{ is a prime number }\}$. However my teacher said that this ideal has finite bases in itself by simple computation. For example: $x_1^2=\frac{1}{2}(x_1^2+x_2^2+x_1^2+x_3^2-x_2^2-x_3^2)$. Maybe he was right, but somehow I just couldn't be persuaded. Hope someone can help. Thanks!
Your ring $R$ certainly has non-Noetherian subrings, for instance $$A=\Bbb C[xy,xy^2,xy^3,\ldots]$$ (I write $x$ for $x_1$ and $y$ for $x_2$). In $A$, the chain of ideals $$(xy)\subseteq(xy,xy^2)\subseteq(xy,xy^2,xy^3)\subseteq\cdots$$ never stabilises.