View source for X-inner automorphism

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 116 formulas, 114 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
$X$-inner automorphisms were introduced by V.K. Kharchenko in [[#References|[a2]]] and [[#References|[a3]]] to study both prime rings satisfying generalized identities and the Galois theory of semi-prime rings (cf. also [[Prime ring|Prime ring]]; [[Rings and algebras|Rings and algebras]]). Since the appropriate definitions are much simpler when the ring is assumed to be prime, this special case is treated first here. Let $R$ be a [[Prime ring|prime ring]] (with $1$) and let $Q = Q _ { s } ( R )$ denote its symmetric [[Martindale ring of quotients|Martindale ring of quotients]]. Then any [[Automorphism|automorphism]] $\sigma$ of $R$ extends uniquely to an automorphism $\hat { \sigma }$ of $Q$, and one says that $\sigma$ is $X$-inner if $\hat { \sigma }$ is inner on $Q$ (cf. also [[Inner automorphism|Inner automorphism]]). It is easy to see that $\operatorname{Inn} ( R )$, the set of all $X$-inner automorphisms of $R$, is a [[Normal subgroup|normal subgroup]] of $\operatorname{Aut}( R )$.

$X$-inner automorphisms control the generalized linear identities of $R$, namely those linear identities which involve automorphisms. For example, it is shown in [[#References|[a2]]] that if $0 \neq a , b , c , d \in R$ and if $\sigma \in \operatorname { Aut } ( R )$ with $a x b = c x ^ { \sigma } d$ for all $x \in R$, then there exists a unit $q \in Q$ with $c = a q$, $d = q ^ { - 1 } b$ and $x ^ { \sigma } = q ^ { - 1 } x q$ for all $x \in R$. In particular, $\hat { \sigma }$ is the inner automorphism of $Q$ induced by $q$ and consequently $\sigma$ is $X$-inner. Of course, $q$ is determined by $\sigma$ up to multiplication by a non-zero element of the extended centroid $C = \mathbf{Z} ( Q ) = \mathbf{C} _ { Q } ( R )$.

Now, let $G$ be a group of automorphisms of $R$ and let $G _ { \operatorname { inn } } = G \cap \operatorname { inn } (R)$, so that $G_{ \operatorname{inn} } \triangleleft G$. If $B ( G )$ denotes the linear span of all units $q$ in $Q$ such that conjugation by $q$ belongs to $\hat { G }_{\text{inn}}$, then $B ( G )$ is a $C$-subalgebra of $Q$, called the algebra of the group (cf. also [[Group algebra|Group algebra]]). One says that $G$ is an $M$-group (Maschke group) if $|G:G_{\operatorname{inn}}|< \infty$ and if $B ( G )$ is a finite-dimensional semi-simple $C$-algebra. Furthermore, $G$ is an $N$-group (Noether group) if $G$ is an $M$-group and if conjugation by every unit of $B ( G )$ induces an automorphism of $R$ contained in $G_{\operatorname{inn}}$. The Galois theory, as developed in [[#References|[a3]]] and [[#References|[a14]]], involves the action of $M$-groups and $N$-groups on prime rings.

Note that if $G$ is an $M$-group and if $q$ is a unit of $B ( G )$, then conjugation by $q$ need not stabilize $R$. Thus, it is not always possible to embed an $M$-group into an $N$-group. One can avoid this difficulty by extending the definition of  "automorphism of R"  to include those (real) automorphisms $\tau$ of $Q$ such that $\tau ( A ) \subseteq R$ for some $0 \neq A \lhd  R$.

$X$-inner automorphisms also appear prominently in the study of cross products. For example, it is proved in [[#References|[a1]]] that if $R  *  G$ is a [[Cross product|cross product]] over the prime ring $R$, then $R  *  G$ embeds naturally into $S = Q ^ { * } G$ and that $\mathbf{C} _ { S } ( R ) = \mathbf{C} _ { S } ( Q )$ is a twisted group algebra $C ^ { t } [ G _ { \text { inn } } ]$ with $Q ^ { * } G _ { \text { inn } } = Q \otimes _ { C } C ^ { t } [ G _ { \text { inn } } ]$. Furthermore, it is shown in [[#References|[a9]]] that every non-zero ideal of $R  *  G$ meets $R ^ { * } G _ { \text { inn } }$ non-trivially, and in [[#References|[a11]]] that $R  *  G$ is prime (or semi-prime) if and only if $R ^ { * } N$ is $G$-prime (or $G$-semi-prime) for all finite normal subgroups $N$ of $G$ contained in $G_{\operatorname{inn}}$. The above-mentioned structure of $R  *  G$ is also used in [[#References|[a5]]] and [[#References|[a6]]] to precisely describe the prime ideals in cross products of finite and of polycyclic-by-finite groups.

There are numerous computations of $\operatorname{Inn} ( R )$ in the literature. To start with, it is shown in [[#References|[a4]]] that if $F$ is a non-commutative free algebra, then $\operatorname { lnn }  ( F ) = \langle 1 \rangle$. More general free products are studied in [[#References|[a7]]] and [[#References|[a8]]]. Next, [[#References|[a10]]] effectively handles graded domains like enveloping algebras of Lie algebras, and [[#References|[a15]]] considers arbitrary enveloping algebra smash products. Finally, [[#References|[a12]]] and [[#References|[a13]]] study certain group algebras and show that for any group $H$ there exists a domain $R$ with $\operatorname { Aut } ( R ) / \operatorname { lnn }  ( R ) \cong H$.

Now suppose that $R$ is a semi-prime ring and again let $Q = Q _ { s } ( R )$ denote its symmetric Martindale ring of quotients. If $\sigma$ is an arbitrary automorphism of $R$, write $\Phi _ { \sigma } = \{ q \in Q : q x ^ { \sigma } = x q \text { for all } x \in R \}$. Then, following [[#References|[a2]]], one says that $\sigma$ is $X$-inner if $\Phi _ { \sigma } \neq 0$. Of course, $\sigma$ is $X$-outer when $\Phi _ { \sigma } = 0$. Note that, in the case of semi-prime rings, $\operatorname{Inn} ( R )$ need not be a subgroup of $\operatorname{Aut}( R )$. Nevertheless, a good deal of structure still exists. For example, [[#References|[a3]]] proves the key fact that $\Phi _ { \sigma }$ is always a cyclic $C$-module.

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  J.W. Fisher,   S. Montgomery,   "Semiprime skew group rings"  ''J. Algebra'' , '''52'''  (1978)  pp. 241–247</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  V.K. Kharchenko,   "Generalized identities with automorphisms"  ''Algebra and Logic'' , '''14'''  (1976)  pp. 132–148  ''Algebra i Logika'' , '''14'''  (1975)  pp. 215–237</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  V.K. Kharchenko,   "Galois theory of semiprime rings"  ''Algebra and Logic'' , '''16'''  (1978)  pp. 208–258  ''Algebra i Logika'' , '''16'''  (1977)  pp. 313–363</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V.K. Kharchenko,   "Algebras of invariants of free algebras"  ''Algebra and Logic'' , '''17'''  (1979)  pp. 316–321  ''Algebra i Logika'' , '''17'''  (1978)  pp. 478–487</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Lorenz,   D.S. Passman,   "Prime ideals in crossed products of finite groups"  ''Israel J. Math.'' , '''33'''  (1979)  pp. 89–132</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top">  M. Lorenz,   D.S. Passman,   "Prime ideals in group algebras of polycyclic-by-finite groups"  ''Proc. London Math. Soc.'' , '''43'''  (1981)  pp. 520–543</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  W.S. Martindale III,   "The normal closure of the coproduct of rings over a division ring"  ''Trans. Amer. Math. Soc.'' , '''293'''  (1986)  pp. 303–317</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  W.S. Martindale III,   S. Montgomery,   "The normal closure of coproducts of domains"  ''J. Algebra'' , '''82'''  (1983)  pp. 1–17</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  S. Montgomery,   "Outer automorphisms of semi-prime rings"  ''J. London Math. Soc.'' , '''18''' :  2  (1978)  pp. 209–220</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  S. Montgomery,   "$X$-inner automorphisms of filtered algebras"  ''Proc. Amer. Math. Soc.'' , '''83'''  (1981)  pp. 263–268</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S. Montgomery,   D.S. Passman,   "Crossed products over prime rings"  ''Israel J. Math.'' , '''31'''  (1978)  pp. 224–256</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  S. Montgomery,   D.S. Passman,   "$X$-Inner automorphisms of group rings"  ''Houston J. Math.'' , '''7'''  (1981)  pp. 395–402</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  S. Montgomery,   D.S. Passman,   "$X$-Inner automorphisms of group rings II"  ''Houston J. Math.'' , '''8'''  (1982)  pp. 537–544</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  S. Montgomery,   D.S. Passman,   "Galois theory of prime rings"  ''J. Pure Appl. Algebra'' , '''31'''  (1984)  pp. 139–184</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  J. Osterburg,   D.S. Passman,   "$X$-inner automorphisms of enveloping rings"  ''J. Algebra'' , '''130'''  (1990)  pp. 412–434</td></tr>
</table>