Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ideal in $R$ with support in $(0,\infty)$. Is $I$ maximal in $R$?



A question arising in the distribution theory of L. Schwartz
The set $J=\{u\in\mathscr D'(\mathbb R):$ supp$u \subseteq [0,\infty)$ and singsupp$u \subseteq (0,\infty)\}$ is a strictly bigger ideal.


To see that it is an ideal decompose such a $u$ by multiplying with a cut-off function as $u=\varphi + v$ where $\varphi \in \mathscr D([0,\infty))$ and $v\in I$. 


