The question ``What kind of rings can be realized as singular cohomology rings of spaces?" is one of the classic problems of algebraic topology posed by Steenrod. This problem, especially for polynomial rings, has been extensively studied by various approaches, with the development of various theories, and finally solved by Anderson, Grodal et al in 2008. In this talk, as a generalization of polynomial rings, we consider the conditions under which the graded Stanley Reisner ring can be expressed as a cohomology ring by using a classical approach.