  ``What kind of ring can be represented as the singular cohomology ring of a
space?'' is a classic problem in algebraic topology, posed by Steenrod. In this
paper, we consider this problem when rings are the graded Stanley-Reisner
rings, in other words the polynomial rings divided by an ideal generated by
square-free monomials. Under some assumption, we give a necessary and
sufficiently condition that a graded Stanley-Reisner ring is realizable.
