Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.

**Question:** Does there exist some character (non-zero multiplicative functional to $\mathbb{C}$) $\omega$ of $\mathcal{S}$ that is *not* the evaluation map at any point in $\mathbb{R}$, i.e. does $\mathcal{S}$ as an algebra, admit characters other than the functionals given by Dirac measures?

Some remarks:

This can not happen if we require $\omega$ to be continuous with respect to the $L^\infty$ norm (one can extend $\omega$ to $C_0(\mathbb{R})$, which is a commutative $C^\ast$-algebra, then use Gelfand-Naimark), my question does not pose any continuity restriction on $\omega$, and can be seen of an algebraic nature.

This amounts to the question whether the ideal $C_c^\infty(\mathbb{R})$ of compactly supported smooth functions is contained in a codimension $1$ ideal of $\mathcal{S}$.