 The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra.Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.Given an abstract simplicial complex Δ on the vertex set {"x"1...,"x""n"} and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k[Δ], is obtained from the polynomial ring k["x"1...,"x""n"] by quotienting out the ideal "I"Δ generated by the square-free monomials corresponding to the non-faces of Δ