An application of the method of orthogonal completeness in graded ring theory. (Q384396)

Using a method of orthogonal completeness developed by \textit{K. I. Beidar} and \textit{A. V. Mikhalev} [Russ. Math. Surv. 40, No. 6, 51-95 (1986); translation from Usp. Mat. Nauk 40, No. 6(246), 79-115 (1985; Zbl 0603.06003)], the author proves a graded analog of a theorem of Herstein. Thus, if \(R\) is a group graded ring which is gr-prime and \(d\) is a homogeneous derivation such that \(d(x)d(y)=d(y)d(x)\) for any \(x,y\in R\), then either \(R\) is commutative or \(d^2=0\). Moreover, a generalization is given for gr-semiprime rings, by showing that a homogeneous derivation extends to a homogeneous derivation of its complete graded right ring of quotients.