Recall that a ring with involution is an algebra \((R;+,\cdot,-,^*,0)\) of type \((2, 2,1,1,0)\) such that \((R;+,\cdot,-,0)\) is an associative ring and for the involution \(^*\) the following identities hold: \((x+y)^*\approx x^* + y^*,(x\cdot y)^* \approx y^*x^*, (x^*)^*\approx x\). A ring with involution is said to have a special involution if it satisfies the identity \(x = xx^*x\). Such rings with involution were studied by M. Yamada. He proved that every ring with special involution satisfies \(x^7 \approx x\). In [Beitr. Algebra Geom. 43, No. 2, 423–432 (2002; Zbl 1016.16021)], S. Crvenković, I. Dolinka and M. Vinčić gave methods for calculating the lattice of rings with involution satisfying \(x^{n+1}\approx x\) for \(n\geq 1\). In particular, they constructed the lattice for \(n = 6\). It has \(90\) elements and is isomorphic to the direct product of a 9-element and a 10-element lattice. The identity base of some varieties of this lattice is well-known, in particular for the varieties generated by subdirectly irreducible rings, but it is unknown for most of its elements.
The main purpose of the paper is to determine the identity bases for these varieties. Moreover, the class of so-called derived rings, which are obtained if we substitute in each ring of such a variety the operations by term operations of the same arity, is studied. The case when the class of derived rings belongs to the original variety is discussed. In particular, the class of derived rings for the variety of rings generated by the two-element Galois-field is described.