Coherence of the ring of periodic distributions

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.