Some algebraic observations on the ring of periodic distributions

Amol Sasane

Published 2017-04-05Version 1

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 ${\mathcal{S}}'({\mathbb{Z}}^d)$ of sequences of at most polynomial growth with termwise operations. In this article we make the following observations: ${\mathcal{S}}'({\mathbb{Z}}^d)$ is not Noetherian; ${\mathcal{S}}'({\mathbb{Z}}^d)$ is a B\'ezout ring; ${\mathcal{S}}'({\mathbb{Z}}^d)$ is coherent; ${\mathcal{S}}'({\mathbb{Z}}^d)$ is a Hermite ring; ${\mathcal{S}}'({\mathbb{Z}}^d)$ is not projective-free; ${\mathcal{S}}'({\mathbb{Z}}^d)$ is a pre-B\'ezout ring; for all $m\in {\mathbb{N}}$, $SL_m({\mathcal{S}}'({\mathbb{Z}}^d))$ is generated by elementary matrices, that is, $SL_m({\mathcal{S}}'({\mathbb{Z}}^d))=E_m({\mathcal{S}}'({\mathbb{Z}}^d))$.