  Let $G$ be a finite group. There is a natural Galois correspondence between
the permutation groups containing $G$ as a regular subgroup, and the Schur
rings (S-rings) over~$G$. The problem we deal with in the paper, is to
characterize those S-rings that are closed under this correspondence, when the
group $G$ is cyclic (the schurity problem for circulant S-rings). It is proved
that up to a natural reduction, the characteristic property of such an S-ring
is to be a certain algebraic fusion of its coset closure introduced and studied
in the paper. Basing on this characterization we show that the schurity problem
is equivalent to the consistency of a modular linear system associated with a
circulant S-ring under consideration. As a byproduct we show that a circulant
S-ring is Galois closed if and only if so is its dual.
