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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, Vojtěchovský-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). [ABSTRACT FROM AUTHOR]

In the two subvarieties of commutative not necessarily associative rings, and 2-divisible not necessarily associative rings, we show that the sixty Bol-Moufang identities determine exactly five subvarieties in each case, and we completely determine all inclusions and necessary counterexamples. The historically important nonassociative rings of octonions and sedenions may be used as counterexamples in our classification. We utilize both associator and linear forms of each Bol-Moufang identity. The results are analogous to the classification of varieties of loops or quasigroups of Bol-Moufang type (Phillips, Vojtechovský-2005). [ABSTRACT FROM AUTHOR]