  In this paper we introduce a new class of polyominoes, called closed paths,
and we study the primality of their associated ideal. Inspired by an existing
conjecture that characterizes the primality of a polyomino ideal by
nonexistence of zig-zag walks, we classify all closed paths which do not
contain zig-zag walks, and we give opportune toric representations of the
associated ideals. To support the conjecture we prove that having no zig-zag
walks is a necessary and sufficient condition for the primality of the
associated ideal of a closed path. Finally, we present some classes of prime
polyominoes viewed as generalizations of closed paths.
