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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]].
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If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
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.
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Out of 49 formulas, 49 were replaced by TEX code.-->
  
<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.
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The obvious Lie analogue of an [[X-inner automorphism|$X$-inner automorphism]].
  
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" />.
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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 derivation $\delta$ of $R$ (cf. also [[Derivation in a ring|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|$X$-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|Prime ring]]), the definition is similar, but considerably more complicated.
  
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.
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$X$-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.
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Next, let $R$ be a prime ring of characteristic $p &gt; 0$ 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|Lie algebra]]) which is a left $C$-vector space, and [[#References|[a3]]] and [[#References|[a4]]] study the Galois theory of $R$ determined by finite-dimensional restricted Lie subrings $L$ of $D ( R )$. Specifically, [[#References|[a3]]] considers the $X$-outer case, where $L$ contains no non-zero inner derivation of $Q$, and [[#References|[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|Quasi-Frobenius ring]]). Note that, if $\operatorname{ad} _ { q } \in L$, then $\operatorname{ad} _ { q }$ is (essentially) an $X$-inner derivation of $R$.
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Finally, $X$-inner derivations appear in [[#References|[a5]]] and [[#References|[a6]]], where the prime ideals of certain enveloping algebra smash products $R \# U ( L )$ are described.
  
 
====References====
 
====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>
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<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>

Latest revision as of 16:46, 1 July 2020

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
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