Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: Theoretical background

We consider the projective line over the finite quotient ring $R_{\diamondsuit}\equiv{GF}(2)[x]/\langle x^3-x\rangle$. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because $R_{\diamondsuit}$ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical $\mathcal J_{\diamondsuit}$ of the ring. The quotient ring $\widetilde R_{\diamondsuit} \equiv R_{\diamondsuit}/\mathcal J_{\diamondsuit}$ is isomorphic to ${GF}(2)\otimes{GF}(2)$. The projective line over $\widetilde R_{\diamondsuit}$ features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper.