  In this paper we introduce some monomial orders for the class of closed path
polyominoes and we prove that the set of the generators of the polyomino ideal
attached to a closed path forms the reduced Gr\"obner basis with respect to
these monomial orders. It is known that the polyomino ideal attached to a
closed path containing an L-configuration or a ladder of at least three steps,
equivalently having no zig-zag walks, is prime. As a consequence, we obtain
that the coordinate ring of a closed path having no zig-zag walks is a normal
Cohen-Macaulay domain.
