In this Wikipedia article about jets, in the section about rigorous definitions, for the algebro-geometric definition, they take the vector space $C^\infty_p(\mathbb R^n,\mathbb R^m)$ of germs of smooth functions at $p$, and then consider the ideal $\mathfrak m_p$ of germs which vanish at $p$ (which they claim to be a maximal ideal of the local ring $C^\infty_p$), then take the quotient $C^\infty_p(\mathbb R^n,\mathbb R^m)/\mathfrak m_p^{k+1}$ and call that a $k$-th order jet space.
My question is: what is the ring structure on this space of germs? I don't see any obvious choice, since functions to $\mathbb R^m$ cannot be multiplied without introducing some additional structure.