# X-inner derivation

The obvious Lie analogue of an $X$-inner automorphism.

Let $R$ be a prime ring (with $1$) and let $Q = Q _ { s } ( R )$ denote its symmetric Martindale ring of quotients. Then any derivation $\delta$ of $R$ (cf. also Derivation in a ring) extends uniquely to a derivation $\hat{\delta}$ of $Q$, and one says that $\delta$ is $X$-inner if $\hat{\delta}$ is inner on $Q$ (cf. also $X$-inner automorphism). It follows easily that $\delta$ is $X$-inner if there exists a $q \in Q$ with $\delta ( x ) = \operatorname { ad } _ { q } ( x ) = [ q , x ]$ for all $x \in R$. Of course, $q$ is determined by $\delta$ up to an additive term in the extended centroid $C = \mathbf{Z} ( Q ) = \mathbf{C} _ { Q } ( R )$. If $R$ is semi-prime (cf. also Prime ring), the definition is similar, but considerably more complicated.

$X$-inner derivations are a tool in the study of differential identities, Galois theory with derivations, and enveloping algebra smash products. To start with, [a1] and [a2] show that the multi-linear differential identities of a semi-prime ring follow from generalized identities, where no derivations are involved, and from certain equations which are always satisfied by derivations. As a consequence, a prime ring satisfying a non-trivial differential identity must also satisfy a non-trivial generalized identity.

Next, let $R$ be a prime ring of characteristic $p > 0$ and let $D ( R )$ denote the set of all derivations $\delta$ of $Q$ such that $\delta ( I _ { \delta } ) \subseteq R$ for some $0 \neq I _ { \delta } \triangleleft R$. Then $D ( R )$ is a restricted Lie ring (cf. also Lie algebra) which is a left $C$-vector space, and [a3] and [a4] study the Galois theory of $R$ determined by finite-dimensional restricted Lie subrings $L$ of $D ( R )$. Specifically, [a3] considers the $X$-outer case, where $L$ contains no non-zero inner derivation of $Q$, and [a4] assumes that $B ( L )$, the $C$-subalgebra of $Q$ generated by all $q \in Q$ with $\operatorname{ad} _ { q } \in L$, is quasi-Frobenius (cf. also Quasi-Frobenius ring). Note that, if $\operatorname{ad} _ { q } \in L$, then $\operatorname{ad} _ { q }$ is (essentially) an $X$-inner derivation of $R$.

Finally, $X$-inner derivations appear in [a5] and [a6], where the prime ideals of certain enveloping algebra smash products $R \# U ( L )$ are described.

How to Cite This Entry:
X-inner derivation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=X-inner_derivation&oldid=50015
This article was adapted from an original article by D.S. Passman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article