Coherence of the ring of periodic distributions

TL;DR

This paper proves the coherence of the ring of periodic distributions and related subrings, analyzing their algebraic properties and relationships with Fourier coefficients, with implications for harmonic analysis.

Contribution

It establishes the coherence of the ring of periodic distributions and its subrings, and explores their algebraic structures such as Hermite and projective free properties.

Findings

The ring of periodic distributions is coherent.

The subring of bounded sequences is coherent.

The subring of convergent sequences is not coherent.

Abstract

It is shown that the ring of periodic distributions is a coherent ring (with the operations of pointwise addition and convolution) by showing that the isomorphic ring $s'$ of the Fourier coefficients (of sequences of at most polynomial growth) with termwise operations is coherent. Moreover, it is shown that the subring $\ell^\infty$ of $s'$ of all bounded sequences is coherent too, while the subring $c$ of $\ell^\infty$ of all convergent sequences is not coherent. It is also observed that $s'$ is a Hermite ring, but not a projective free ring.