  In this work, we consider a family of coordinate rings obtained as a quotient
ring of a binomial ideal associated to a collection of cells. The algebraic
properties of Cohen-Macaulayness, Gorensteiness and the Hilbert-Poincar\'{e}
series have been studied in literature only for some classes of collections of
cells whose related binomial ideals are prime. We study these algebraic
properties for some classes of collections of cells and polyominoes whose
coordinate rings are not domain, considering a new strategy involving the
combinatorics of the collections of cells and a recent result of Conca and
Varbaro on the radical initial ideals.
