Consider the ring of functions $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ from $\mathbb{R}^d$ to itself with $k$-continuous derivatives. Is the localization of $C^{k}(\mathbb{R}^d;\mathbb{R}^d)$ at the ideal $(0)$ a field?
If so, is this object ever studied?