Is the given Ring $\mathcal K(U)$ an integral domain?

Question

Let $U$ be a bounded open disk in $\mathbb C$ and $\mathcal K(U)$ denote the ring of complex analytic function on $U$.

1. Is $\mathcal K(U)$ an integral domain .

2. Give an example of a maximal ideal in $\mathcal K(U)$.

In first part we need only to show that $\mathcal K(U)$ is a without zero divisor.

Please give me hint how to show that this ring is a without zero divisor.

Thank you.

Answer 1

Hint: complex analytic functions are locally power series. Show that the ring of formal power series over $\Bbb C$ has no zero divisors.