Ultrafilters/prime ideals in a Boolean algebra

Question

Let $U$ be an ultrafilter in a Boolean algebra $A$. Does $U$ necessarily contains an atom of $A$?

Let $I$ be a prime ideal of the same Boolean algebra $A$. Does $I$ necessarily contains the complement of an atom of $A$?

Edit: Sorry, I meant in a finite Boolean algebra

Answer 1

If $X$ is an infinite set, and $I$ is the ideal or $\wp(X)$ consisting of the finite subsets of $X$, then the Boolean algebra $\wp(X)/I$ is atomless. Other atomless Boolean algebras include the Boolean algebra of regular open subsets of $\Bbb R^n$ and the interval algebra of the real numbers.