View source for X-inner derivation

The obvious Lie analogue of an [[X-inner automorphism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200202.png" />-inner automorphism]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200203.png" /> be a [[Prime ring|prime ring]] (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200204.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200205.png" /> denote its symmetric [[Martindale ring of quotients|Martindale ring of quotients]]. Then any derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200207.png" /> (cf. also [[Derivation in a ring|Derivation in a ring]]) extends uniquely to a derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200208.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x1200209.png" />, and one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002012.png" />-inner if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002013.png" /> is inner on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002014.png" /> (cf. also [[X-inner automorphism|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002015.png" />-inner automorphism]]). It follows easily that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002017.png" />-inner if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002020.png" />. Of course, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002021.png" /> is determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002022.png" /> up to an additive term in the extended centroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002024.png" /> is semi-prime (cf. also [[Prime ring|Prime ring]]), the definition is similar, but considerably more complicated.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002025.png" />-inner derivations are a tool in the study of differential identities, Galois theory with derivations, and enveloping algebra smash products. To start with, [[#References|[a1]]] and [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002026.png" /> be a prime ring of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002027.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002028.png" /> denote the set of all derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002031.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002032.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002033.png" /> is a restricted Lie ring (cf. also [[Lie algebra|Lie algebra]]) which is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002034.png" />-vector space, and [[#References|[a3]]] and [[#References|[a4]]] study the Galois theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002035.png" /> determined by finite-dimensional restricted Lie subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002037.png" />. Specifically, [[#References|[a3]]] considers the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002038.png" />-outer case, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002039.png" /> contains no non-zero inner derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002040.png" />, and [[#References|[a4]]] assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002041.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002042.png" />-subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002043.png" /> generated by all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002045.png" />, is quasi-Frobenius (cf. also [[Quasi-Frobenius ring|Quasi-Frobenius ring]]). Note that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002047.png" /> is (essentially) an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002048.png" />-inner derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002049.png" />.

Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002050.png" />-inner derivations appear in [[#References|[a5]]] and [[#References|[a6]]], where the prime ideals of certain enveloping algebra smash products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120020/x12002051.png" /> are described.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.K. Kharchenko,   "Differential identities of prime rings"  ''Algebra and Logic'' , '''17'''  (1979)  pp. 155–168  ''Algebra i Logika'' , '''17'''  (1978)  pp. 220–238</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.K. Kharchenko,   "Differential identities of semiprime rings"  ''Algebra and Logic'' , '''18'''  (1979)  pp. 58–80  ''Algebra i Logika'' , '''18'''  (1979)  pp. 86–119</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.S. Passman,   "Prime ideals in enveloping rings"  ''Trans. Amer. Math. Soc.'' , '''302'''  (1987)  pp. 535–560</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.S. Passman,   "Prime ideals in restricted enveloping rings"  ''Commun. Algebra'' , '''16'''  (1988)  pp. 1411–1436</TD></TR></table>