  Motivated by the Schwartz space of tempered distributions $\mathscr S^\prime$
and the Kondratiev space of stochastic distributions $\mathcal S_{-1}$ we
define a wide family of nuclear spaces which are increasing unions of (duals
of) Hilbert spaces $\mathscr H_p^\prime,p\in\mathbb N$, with decreasing norms
$|\cdot|_{p}$. The elements of these spaces are functions on a free commutative
monoid. We characterize those rings in this family which satisfy an inequality
of the form $|f * g|_{p} \leq A(p-q) |f|_{q}|g|_{p}$ for all $p\ge q+d$, where
* denotes the convolution in the monoid, $A(p-q)$ is a strictly positive number
and $d$ is a fixed natural number (in this case we obtain commutative
topological $\mathbb C$-algebras). Such an inequality holds in $\mathcal
S_{-1}$, but not in $\mathscr S^\prime$. We give an example of such a ring
which contains $\mathscr S^\prime$. We characterize invertible elements in
these rings and present applications to linear system theory
