The most general way I can formulate my question is the following:

Question 1: *Given a Gorenstein quotient ring $S$ of a polynomial ring over a field $K$, can one construct a (topological) space $X$ such that the (even degree part of the) singular cohomology ring of $X$ with coefficients in $K$ is isomorphic to $S$?*

Edit: as mentioned in the comments, the answer depends on the grading of the variables. I would be most interested in a uniform grading (i.e all variables have the same degree d, without restriction on the value of d).

For the specific cases I have in mind, $S$ is a quotient of a polynomial ring over $\mathbb{C}$ by an ideal generated by monomials and binomials. Below is an example of such a ring $S$ that I would like to realize as the cohomology ring of some space:

$$ S = \mathbb{C}[x_1,\ldots,x_7]/(x_7^2, x_3x_7-x_4x_7, x_2x_7-x_5x_7, x_1x_7-x_6x_7, x_6^2, x_3x_6-x_5x_6, x_2x_6-x_4x_6, x_1x_6-x_6x_7, x_5^2, x_3x_5-x_5x_6, x_2x_5-x_5x_7, x_1x_5-x_4x_5, x_4^2, x_3x_4-x_4x_7, x_2x_4-x_4x_6, x_1x_4-x_4x_5, x_3^2, x_1x_3-x_2x_3, x_2^2, x_1x_2-x_2x_3, x_1^2).$$

The ring $S$ above is Gorenstein and all rings that I am considering for my question are also Gorenstein (i.e satisfy Poincare duality).

Edit 2: I will put this here rather than leaving the additional question in the comments because it goes deeper into what I was really interested in.

Question 2: *Since the answer to Question 1 seems to be affirmative, can the space X be chosen to be "nice" (e.g. compact manifold) under the assumption that $S$ (the cohomology ring of $X$) satisfies Poincare duality?*