The Kleinfeld identities in generalized accessible rings

It is proved that the identities ([x, y]⁴, z, t) = ([x, y]², z, t) [x, y] = [x, y] ([x, y]², z, t) = 0, known in the theory of alternative rings as the Kleinfeld identities, are fulfilled in every generalized accessible ring of characteristic different from 2 and 3. These identities allow us to construct central and kernel functions in the variety of generalized accessible rings. It is also proved that in a free generalized accessible and a free alternative ring with more than three generators the Kleinfeld element ([x, y]², z, t) is nilpotent of index 2.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 291–297, February, 1976.