If $W$ is a proper closed subset of $V$, then $W$ correspondes to a prime ideal.

Question

Let $V$ be a $k$-variety $1$-dimensional. If $W$ is a proper closed subset of $V$, then $W$ correspondes to a prime ideal?

I saw this statement being used in a demonstration, but I couldn't prove it. Any hint?

Answer 1

That's wrong. The ideal will be prime iff $W$ is a reduced point, so for example two points give a non-prime ideal.