An application of the method of orthogonal completeness in graded ring theory

A method of orthogonal completeness was devised by K. I. Beidar and A. V. Mikhalev in the 1970s. Initially, the method was applied in ring theory and was mainly used to derive theorems for semiprime rings by reducing the semiprime case to the prime. In the 1980s, the same authors developed a theory of orthogonal completeness for arbitrary algebraic systems. The theory of orthogonal completeness is applied to group-graded rings. To use the Beidar–Mikhalev theorems on orthogonal completeness, a graded ring is treated as an algebraic system with a ring signature augmented by the operation of taking homogeneous components and by homogeneity predicates. The graded analog of Herstein's theorem for prime rings with derivations, as well as its generalization to semiprime rings based on the method of orthogonal completeness, is proved. It is shown that every homogeneous derivation of a graded ring extends to a homogeneous derivation of its complete graded right ring of quotients.