Differential identities, Lie ideals, and Posner’s theorems.

*(English)*Zbl 0614.16028This paper uses the theory of differential identities to obtain generalizations of two well-known results of E. C. Posner [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)]. A number of such generalizations appear in the literature and the purpose here is to give a uniform treatment which yields essentially all of these, and gives new results as well. Let \(R\) be a prime ring with center \(Z\), extended centroid \(C\), and noncentral Lie ideal \(L\). When \(R\) has an involution, \(^ *\), let \(I=I^ *\) be a nonzero ideal of \(R\) and denote the symmetric and skew-symmetric elements of \(I\) by \(T(I)\) and \(K(I)\), respectively.

The first part of the paper shows that if \(d\) is a nonzero derivation of \(R\) so that \([x,x^ d]\in Z\) for all \(x\in L\), then either \(R\) is commutative, or char \(R=2\) and \(R\) satisfies the standard identity \(S_ 4\). Furthermore, if \(R\) has an involution and this same relation holds for all \(x\in T(I)\), or all \(x\in K(I)\), then \(R\) satisfies \(S_ 4\). The generalization of Posner’s other theorem is that if \(d\) and \(h\) are nonzero derivations of \(R\) so that \(dh\) is a Lie derivation of \(L\) into \(R\) then char \(R=2\) and either \(R\) satisfies \(S_ 4\), or \(h=dc\) for \(c\in C\). Similar, but less definitive results are obtained when one assumes that \(R\) has an involution and that \(dh\) is either a Lie derivation of \(K(I)\) to \(R\), or a Jordan derivation of \(T(I)\) to \(R\).

The first part of the paper shows that if \(d\) is a nonzero derivation of \(R\) so that \([x,x^ d]\in Z\) for all \(x\in L\), then either \(R\) is commutative, or char \(R=2\) and \(R\) satisfies the standard identity \(S_ 4\). Furthermore, if \(R\) has an involution and this same relation holds for all \(x\in T(I)\), or all \(x\in K(I)\), then \(R\) satisfies \(S_ 4\). The generalization of Posner’s other theorem is that if \(d\) and \(h\) are nonzero derivations of \(R\) so that \(dh\) is a Lie derivation of \(L\) into \(R\) then char \(R=2\) and either \(R\) satisfies \(S_ 4\), or \(h=dc\) for \(c\in C\). Similar, but less definitive results are obtained when one assumes that \(R\) has an involution and that \(dh\) is either a Lie derivation of \(K(I)\) to \(R\), or a Jordan derivation of \(T(I)\) to \(R\).

##### MSC:

16W20 | Automorphisms and endomorphisms |

16Rxx | Rings with polynomial identity |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |

16U70 | Center, normalizer (invariant elements) (associative rings and algebras) |

16N60 | Prime and semiprime associative rings |