Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: The Mermin "magic" square/pentagram

In 1993, Mermin gave surprisingly simple proofs of the Bell–Kochen–Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin–Peres "magic" square and the Mermin pentagram. The former is a $3\times 3$ array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin–Peres square and the points of the projective line over the product ring $GF(2)\otimes GF(2)$. Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column's observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over $GF(2)[x]/\langle x^3-x\rangle$. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over $GF(2)$, their three "Jacobson" counterparts, and the four points whose both coordinates are zero divisors. The other configuration features the neighborhood of the point $(1,0)$ (or, equivalently, that of $(0,1)$). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions.