Given the ring $R:=\{f:\mathbb{R}\rightarrow \mathbb{R}, f $ is continous$\}$ we define for $c,d \in \mathbb{R}$ where $c\neq d$ , the ideal $I_{c,d}:=\{f\in R|f(c)=f(d)=0\} \subset R$. I have to show that $I_{c,d}$ is not a prime ideal.
My attempt is to find a counterexample:
Let $R$ be $\mathbb{R}[x]$. Then $f=(x-c)(x-d)$ with $f \in I_{c,d}$ but $(x-c) \notin I_{c,d}$ and $(x-d) \notin I_{c,d}$. So it's not a prime ideal. Is this enough? Or is there a better way to show it? Thanks for help.
Yes it is a correct answer, when solving this kind of problem you can also consider the idea of taking the quotien ring and check whether or not it's a domain.