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.

References

[a1]	V.K. Kharchenko, "Differential identities of prime rings" Algebra and Logic , 17 (1979) pp. 155–168 Algebra i Logika , 17 (1978) pp. 220–238
[a2]	V.K. Kharchenko, "Differential identities of semiprime rings" Algebra and Logic , 18 (1979) pp. 58–80 Algebra i Logika , 18 (1979) pp. 86–119
[a3]	V.K. Kharchenko, "Constants of derivations of prime rings" Math. USSR Izv. , 45 (1982) pp. 381–401 Izv. Akad. Nauk SSSR Ser. Mat. , 45 (1981) pp. 435–461
[a4]	V.K. Kharchenko, "Derivations of prime rings of positive characteristic" Algebra and Logic , 35 (1996) pp. 49–58 Algebra i Logika , 35 (1996) pp. 88–104
[a5]	D.S. Passman, "Prime ideals in enveloping rings" Trans. Amer. Math. Soc. , 302 (1987) pp. 535–560
[a6]	D.S. Passman, "Prime ideals in restricted enveloping rings" Commun. Algebra , 16 (1988) pp. 1411–1436