Proof that $\mathbb{C}[x,y]$ is not a semisimple ring?

Question

The title says it all ($\mathbb C$ denotes the complex numbers). Perhaps it is not Artinian but I don't know how to prove that either... Any help would be much appreciated

Answer 1

Let $f(x,y)\in\mathbb{C}[x,y]$ have positive (total) degree. Then $$ (f)\supsetneq (f^2)\supsetneq (f^3)\supsetneq\dotsb $$

Alternatively, a commutative semisimple ring is a product of fields, by the Artin-Wedderburn theorem. In particular, an integral domain is semisimple if and only if it is a field.

Answer 2

Another way to see the difference:

In a semisimple ring, every right ideal is generated by an idempotent. In fact, every element is the product of an idempotent and unit.

On the other hand, a domain like $\mathbb C[x,y]$ only has two idempotents: $\{0,1\}$.