A functional identity with an automorphism in semiprime rings (Q2762290)

Let \(R\) be a semiprime ring, \(Q\) its left Martindale quotient ring, and \(C\) the extended centroid of \(R\). If \(\sigma\) is an automorphism of \(R\), \(S\) is a nonempty set, \(f,g,h,k\colon S\to R\) are functions, and for all \(x\in R\) and \(s,t\in S\), \(f(s)xg(t)=h(s)\sigma(x)k(t)\), the author proves that there are orthogonal idempotents \(e_1,\dots,e_5\in C\) with sum \(1\) in \(Q\) so that \(e_1\sigma(x)=e_1qxq^{-1}\), \(e_1f(s)=e_1h(s)q\) and \(e_1g(s)=e_1q^{-1}k(s)\), and each of the other \(e_i\) annihilates a pair including one of \(f\) or \(g\) and one of \(h\) or \(k\), e.g. \(e_2g(s)=0=e_2 k(s)\). This result is used to give a similar characterization of \(\sigma\)-biderivations \(\Delta\) of \(R\); \(e_1\sigma\) is still inner, \(e_1\Delta\) is essentially the usual Lie commutator, \(e_2\Delta=0\), and \(e_3R\) is commutative.