A potpourri of algebraic properties of the ring of periodic distributions

The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with termwise operations. In this article, we establish several algebraic properties of these rings.