  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.
