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Differential identities, Lie ideals, and Posner’s theorems. (English) Zbl 0614.16028
This 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 $$R$$.

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
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