Stalk of a point in the intersection of two irreducible components of a Noetherian scheme

Question

If we have a Noetherian scheme $X$, is it true that for any point $p$ that is in two irreducible components of $X$, then the stalk of $X$ at $p$ is not an integral domain?

Answer 1

The minimal primes of $\mathscr{O}_{X,p}$ are in canonical bijection with the irreducible components of $X$ passing through $p$. So, if there are two components passing through $p$, $\mathscr{O}_{X,p}$ has at least two minimal primes, and therefore cannot be a domain.

This is true whether or not $X$ is Noetherian.