# Ideals of a $C^{\infty}$-ring

Let $$M$$ be a $$n$$-dimensional smooth manifold and $$F:M\rightarrow\mathbb{R}^{k}$$ a smooth map. If $$0\in\mathbb{R}^{k}$$ is a regular value of $$F$$, then the level set $$N:=F^{-1}(0)$$ is a smooth submanifold in $$M$$ with dimension $$n-k$$. Moreover, a standard result in differential geometry states that the tangent space $$T_{p}N$$ ($$p\in N$$) is equal to the kernel of the tangent map at $$p$$, i.e. $$T_{p}N=\ker\,(dF_{p}:T_{p}M\rightarrow T_{0}\mathbb{R}^{k}\cong\mathbb{R}^{k}).$$

If we drop the regular condition, then the level set $$N=F^{-1}(0)$$ is not a manifold. However, we can define a "smooth structure" on $$N$$ as follows. Given an open subset $$U$$ in $$M$$, then $$U\cap N$$ is open in $$N$$. We say that a function $$h:U\cap N\rightarrow\mathbb{R}$$ is smooth, if $$h=H|_{U\cap N}$$ for some smooth function $$H$$ on $$U$$. Denote by $$C^{\infty}_{N}$$ the sheaf of smooth function on $$N$$. We get a ringed space $$(N,C^{\infty}_{N})$$. In fact, the stalks of $$C^{\infty}_{N}$$ are locally $$C^{\infty}$$-rings and therefore $$(N,C^{\infty}_{N})$$ is a locally $$C^{\infty}$$-ringed space(cf. Definition 4.8 in "Algebraic Geometry over $$C^{\infty}$$-Rings" by D. Joyce). Given a point $$p$$ in $$N$$, let $$\mathfrak{m}_{p}$$ be the maximal ideal of $$C^{\infty}_{N,p}$$. The Zariski tangent space of $$N$$ at $$p$$ is $$T^{Zar}_{p}N=\mathrm{Hom}_{C^{\infty}_{N,p}}(\mathfrak{m}_{p}/\mathfrak{m}^{2}_{p},C^{\infty}_{N,p}) \cong\mathrm{Der}_{\mathbb{R}}(C^{\infty}_{N,p},C^{\infty}_{N,p}).$$ My question is: does the equality $$T^{Zar}_{p}N\cong\ker\,(dF_{p}:T_{p}M\rightarrow T_{0}\mathbb{R}^{k}\cong\mathbb{R}^{k})$$ still hold? One direction is obvious, $$T^{Zar}_{p}N$$, as a subspace of $$T_{p}M$$, is contained in $$\ker\,(dF_{p})$$. To prove the assertion, it suffices to show that each vector in $$\ker\,(dF_{p})$$ (considered as a derivation on $$C^{\infty}_{M,p}$$) vanishes on the ideal $$J_{p}=\{[h]_{p}\in C^{\infty}_{N,p}\,|h|_{N}=0\}.$$ By contrast to the conventional algebraic geometry, the local $$C^{\infty}$$-ring $$C^{\infty}_{N,p}$$ is not Noetherian. I don't know how to characterize the idea $$J_{p}$$ explicitly.