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Motivated by the Schwartz space of tempered distributions $\mathscr S^\prime$and the Kondratiev space of stochastic distributions $\mathcal S_{-1}$ wedefine a wide family of nuclear spaces which are increasing unions of (dualsof) 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 commutativemonoid. We characterize those rings in this family which satisfy an inequalityof 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 numberand $d$ is a fixed natural number (in this case we obtain commutativetopological $\mathbb C$-algebras). Such an inequality holds in $\mathcalS_{-1}$, but not in $\mathscr S^\prime$. We give an example of such a ringwhich contains $\mathscr S^\prime$. We characterize invertible elements inthese rings and present applications to linear system theory

New Topological C-Algebras with Applications in Linear Systems Theory