The varieties of rings of Bol-Moufang type

In the variety of not necessarily associative rings, we show that the sixty Bol-Moufang identities determine exactly thirteen subvarieties, and we determine all inclusions and necessary counterexamples. We make use of the concepts of magma rings and linear identities. The result is analogous to the classification of varieties of loops or quasigroups of Bol-Moufang type (Phillips, Vojtechovsky-2005) and was motivated by the study of tangent algebras to smooth loops of Bol-Moufang type (Mal'cev-1955, Sagle-1961, Nagy 1992, Bremner, Madariaga-2014).