Let $I$ be an ideal of the ring of smooth functions $\mathcal{C}^{\infty}(\mathbb{R})$ defined by $$I=\left\{f\in\mathcal{C}^{\infty}(\mathbb{R})\,:\,f^{(n)}(0)=0~\textrm{for all $n\in\mathbb{Z}_{\ge0}$}\right\}.$$ Then, $I$ is a prime ideal of $\mathcal{C}^{\infty}(\mathbb{R})$.
What is the isomorphic image of the quotient ring $\mathcal{C}^{\infty}(\mathbb{R})/I$?
Which domain can be considerd as an isomorphic image of $\mathcal{C}^{\infty}(\mathbb{R})/I$?