Centralizing mappings and derivations in prime rings
Let \(R\) be a prime ring with center \(Z\) and extended centroid \(C\). For \(x,y\in R\) set \([x,y]=xy-yx\). An additive endomorphism \(F\) of \(R\) is called centralizing on \(A\subseteq R\) if \([a,F(a)]\in Z\) for all \(a\in A\), and is called commuting if \([a,F(a)]=0\) for all \(a\in A\). The first main result of this interesting paper characterizes a centralizing mapping \(F\) by showing that \(F(x)=\lambda x+\mu(x)\) for \(\lambda\in Z\) and \(\mu: (R,+)\to (C,+)\) whenever \(\text{char}(R)\neq 2\) or \(F\) is commuting. This result requires the following: Theorem. If \(D\), \(G\), and \(H\) are derivations of \(R\), \(a,b\in R-Z\), and \(D(x)=aG(x)+H(x)b\) for all \(x\in R\), then for some \(\lambda\in C\), \(D(x)=[\lambda ab,x]\), \(G(x)=[\lambda b,x]\), and \(H(x)=[\lambda a,x]\). The second main result of the paper, which has the most involved proof, is: Theorem. If \(D\) and \(H\) are derivations of \(R\) so that \(D(u)u-uH(u)\in Z\) for all \(u\in U\), a nonzero left ideal of \(R\), then either \(D=0\) or \(R\) is commutative. The proofs are computational, but are well presented and easy to follow.
scientific article