Smooth functions on preimage of a regular value

Question

Let $M$ be a smooth manifold, and $f:M\to \mathbb{R}$ a smooth function, such that $0\in \mathbb{R}$ is a regular value. Then $f^{-1}(0)$ is a smooth submanifold, and we get an algebra map $$ f^\ast:C^\infty(M)/(f) \to C^\infty(f^{-1}(0)) $$ defined by $f^\ast(g+(f)) = \left.g\right|_{f^{-1}(0)}$. My question was whether this map is an isomorphism. It seems to me that this should be the case, so I was wondering if someone could help me out here. Thanks in advance!