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In this paper we introduce some monomial orders for the class of closed pathpolyominoes and we prove that the set of the generators of the polyomino idealattached to a closed path forms the reduced Gr\"obner basis with respect tothese monomial orders. It is known that the polyomino ideal attached to aclosed 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 obtainthat the coordinate ring of a closed path having no zig-zag walks is a normalCohen-Macaulay domain.

On Gr\"obner bases and Cohen-Macaulay property of closed path polyominoes