On some geometrical properties defining classes of rings and varieties of modular lattices (Q791552)

Several authors have considered the question of translating axiomatic geometry into the language of lattice theory. For example, it is well known that the modular law distinguishes projective geometries, and Jónsson's law distinguishes desarguean projective geometries. In a previous publication, the author has considered two identities that reflect Pappus' theorem. In the present paper he considers these pappian identities more closely. He uses what he calls the quasi-plane identity to solve the problem of Skornyakov: to find a lattice equation which, when applied to \({\mathcal L}(_ RR^ 3)\) for a regular ring R, characterises the commutativity of the ring.