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''of a ring''
 
''of a ring''
  
The largest quasi-regular ideal of the given ring. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766701.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766702.png" /> is called quasi-regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766703.png" /> is a [[Quasi-regular ring|quasi-regular ring]]. There exists in every alternative (in particular, associative) ring a quasi-regular radical; it coincides with the sum of all right (left) quasi-regular ideals (see [[#References|[1]]], [[#References|[10]]]). The quasi-regular radical of an associative ring is also called the [[Jacobson radical|Jacobson radical]].
+
The largest quasi-regular ideal of the given ring. An ideal $  A $
 +
of a ring $  R $
 +
is called quasi-regular if $  A $
 +
is a [[Quasi-regular ring|quasi-regular ring]]. There exists in every alternative (in particular, associative) ring a quasi-regular radical; it coincides with the sum of all right (left) quasi-regular ideals (see [[#References|[1]]], [[#References|[10]]]). The quasi-regular radical of an associative ring is also called the [[Jacobson radical|Jacobson radical]].
  
The quasi-regular radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766704.png" /> of an arbitrary alternative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766705.png" /> is equal to the intersection of all maximal modular right (left) ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766706.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766707.png" /> is also equal to the intersection of the kernels of all irreducible right (left) representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766708.png" /> (see [[#References|[1]]], [[#References|[5]]]–[[#References|[8]]]). A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q0766709.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667011.png" />-semi-simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667012.png" />. The quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667013.png" /> is always semi-simple. Every [[Semi-simple ring|semi-simple ring]] is isomorphic to a subdirect sum of primitive rings [[#References|[1]]], [[#References|[8]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667014.png" /> satisfies the minimum condition for right (left) ideals, then the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667015.png" /> is nilpotent and the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667016.png" /> is isomorphic to a finite direct sum of complete matrix rings over skew-fields and Cayley–Dickson algebras (the latter terms are absent in the associative case), see [[#References|[1]]]–[[#References|[3]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667017.png" /> be a two-sided ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667018.png" />; then
+
The quasi-regular radical $  J ( R) $
 +
of an arbitrary alternative ring $  R $
 +
is equal to the intersection of all maximal modular right (left) ideals of $  R $;  
 +
$  J ( R) $
 +
is also equal to the intersection of the kernels of all irreducible right (left) representations of $  R $(
 +
see [[#References|[1]]], [[#References|[5]]]–[[#References|[8]]]). A ring $  R $
 +
is called $  J $-
 +
semi-simple if $  J ( R) = 0 $.  
 +
The quotient ring $  R / J ( R) $
 +
is always semi-simple. Every [[Semi-simple ring|semi-simple ring]] is isomorphic to a subdirect sum of primitive rings [[#References|[1]]], [[#References|[8]]]. If $  R $
 +
satisfies the minimum condition for right (left) ideals, then the radical $  J ( R) $
 +
is nilpotent and the quotient ring $  R / J ( R) $
 +
is isomorphic to a finite direct sum of complete matrix rings over skew-fields and Cayley–Dickson algebras (the latter terms are absent in the associative case), see [[#References|[1]]]–[[#References|[3]]]. Let $  A $
 +
be a two-sided ideal of the ring $  R $;  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667019.png" /></td> </tr></table>
+
$$
 +
J ( A)  = A \cap J ( R)
 +
$$
  
(see [[#References|[1]]], [[#References|[4]]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667020.png" /> is an associative ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667021.png" /> is the ring of matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667023.png" />, then
+
(see [[#References|[1]]], [[#References|[4]]]); if $  R $
 +
is an associative ring and $  R _ {n} $
 +
is the ring of matrices of order $  n $
 +
over $  R $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667024.png" /></td> </tr></table>
+
$$
 +
J ( R _ {n} )  = [ J ( R) ] _ {n} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667025.png" /> is an associative algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667026.png" /> and the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667027.png" /> is greater than the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667028.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667029.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667030.png" /> is algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667032.png" /> is a nil ideal. A quasi-regular radical of a finitely-generated alternative ring satisfying an essential identity relation is the same as a lower nil radical (see [[Radical of rings and algebras|Radical of rings and algebras]]) [[#References|[6]]]. Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. [[Jordan algebra|Jordan algebra]]).
+
If $  R $
 +
is an associative algebra over a field $  F $
 +
and the cardinality of $  F $
 +
is greater than the dimension of $  R $
 +
over $  F $
 +
or if $  R $
 +
is algebraic over $  F $,  
 +
then $  J ( R) $
 +
is a nil ideal. A quasi-regular radical of a finitely-generated alternative ring satisfying an essential identity relation is the same as a lower nil radical (see [[Radical of rings and algebras|Radical of rings and algebras]]) [[#References|[6]]]. Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. [[Jordan algebra|Jordan algebra]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.A. Zhevlakov,  "On radical ideals of an alternative ring"  ''Algebra i Logika'' , '''4''' :  4  (1965)  pp. 87–102  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,  "Alternative Artinian rings"  ''Algebra i Logika'' , '''5''' :  3  (1966)  pp. 11–36  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K.A. Zhevlakov,  "On the Kleinfeld and Smiley radicals of alternative rings"  ''Algebra and Logic'' , '''8''' :  2  (1969)  pp. 100–102  ''Algebra i Logika'' , '''8''' :  2  (1969)  pp. 176–180</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.A. Zhevlakov,  "Coincidence of Kleinfeld and Smiley radicals in alternative rings"  ''Algebra and Logic'' , '''8''' :  3  (1969)  pp. 175–181  ''Algebra i Logika'' , '''8''' :  3  (1969)  pp. 309–319</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K.A. Zhevlakov,  "Quasiregular ideals in finitely generated alternative rings"  ''Algebra and Logic'' , '''11''' :  2  (1972)  pp. 94–100  ''Algebra i Logika'' , '''11''' :  2  (1972)  pp. 140–1161</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.M. Slin'ko,  I.P. Shestakov,  "Right representation of algebras"  ''Algebra and Logic'' , '''13''' :  5  (1973)  pp. 312–333  ''Algebra i Logika'' , '''13''' :  5  (1974)  pp. 544–588</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Kleinfeld,  "Primitive alternative rings and semi-simplicity"  ''Amer. J. Math.'' , '''77'''  (1955)  pp. 725–730</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  K. McCrimmon,  "The radical of a Jordan algebra"  ''Proc. Nat. Acad. Sci. USA'' , '''62'''  (1969)  pp. 671–678</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.F. Smiley,  "The radical of an alternative ring"  ''Ann. of Math.'' , '''49''' :  3  (1948)  pp. 702–709</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.A. Zhevlakov,  "On radical ideals of an alternative ring"  ''Algebra i Logika'' , '''4''' :  4  (1965)  pp. 87–102  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,  "Alternative Artinian rings"  ''Algebra i Logika'' , '''5''' :  3  (1966)  pp. 11–36  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K.A. Zhevlakov,  "On the Kleinfeld and Smiley radicals of alternative rings"  ''Algebra and Logic'' , '''8''' :  2  (1969)  pp. 100–102  ''Algebra i Logika'' , '''8''' :  2  (1969)  pp. 176–180</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.A. Zhevlakov,  "Coincidence of Kleinfeld and Smiley radicals in alternative rings"  ''Algebra and Logic'' , '''8''' :  3  (1969)  pp. 175–181  ''Algebra i Logika'' , '''8''' :  3  (1969)  pp. 309–319</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K.A. Zhevlakov,  "Quasiregular ideals in finitely generated alternative rings"  ''Algebra and Logic'' , '''11''' :  2  (1972)  pp. 94–100  ''Algebra i Logika'' , '''11''' :  2  (1972)  pp. 140–1161</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.M. Slin'ko,  I.P. Shestakov,  "Right representation of algebras"  ''Algebra and Logic'' , '''13''' :  5  (1973)  pp. 312–333  ''Algebra i Logika'' , '''13''' :  5  (1974)  pp. 544–588</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Kleinfeld,  "Primitive alternative rings and semi-simplicity"  ''Amer. J. Math.'' , '''77'''  (1955)  pp. 725–730</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  K. McCrimmon,  "The radical of a Jordan algebra"  ''Proc. Nat. Acad. Sci. USA'' , '''62'''  (1969)  pp. 671–678</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.F. Smiley,  "The radical of an alternative ring"  ''Ann. of Math.'' , '''49''' :  3  (1948)  pp. 702–709</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The radical of a row-finite infinite matrix ring is determined by annihilating sequences of ideals [[#References|[a1]]]. The radical of a polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667033.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667034.png" /> for some nil ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667036.png" /> (the determination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076670/q07667037.png" /> is an open problem).
+
The radical of a row-finite infinite matrix ring is determined by annihilating sequences of ideals [[#References|[a1]]]. The radical of a polynomial ring $  R [ x] $
 +
is $  N [ x] $
 +
for some nil ideal $  N $
 +
in $  R $(
 +
the determination of $  N $
 +
is an open problem).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Sexauer,  J.E. Warnock,  "The radical of the row-finite matrices over an arbitrary ring"  ''Trans. Amer. Math. Soc.'' , '''39'''  (1969)  pp. 281–295</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I, II''' , Acad. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Sexauer,  J.E. Warnock,  "The radical of the row-finite matrices over an arbitrary ring"  ''Trans. Amer. Math. Soc.'' , '''39'''  (1969)  pp. 281–295</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I, II''' , Acad. Press  (1988)</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


of a ring

The largest quasi-regular ideal of the given ring. An ideal $ A $ of a ring $ R $ is called quasi-regular if $ A $ is a quasi-regular ring. There exists in every alternative (in particular, associative) ring a quasi-regular radical; it coincides with the sum of all right (left) quasi-regular ideals (see [1], [10]). The quasi-regular radical of an associative ring is also called the Jacobson radical.

The quasi-regular radical $ J ( R) $ of an arbitrary alternative ring $ R $ is equal to the intersection of all maximal modular right (left) ideals of $ R $; $ J ( R) $ is also equal to the intersection of the kernels of all irreducible right (left) representations of $ R $( see [1], [5][8]). A ring $ R $ is called $ J $- semi-simple if $ J ( R) = 0 $. The quotient ring $ R / J ( R) $ is always semi-simple. Every semi-simple ring is isomorphic to a subdirect sum of primitive rings [1], [8]. If $ R $ satisfies the minimum condition for right (left) ideals, then the radical $ J ( R) $ is nilpotent and the quotient ring $ R / J ( R) $ is isomorphic to a finite direct sum of complete matrix rings over skew-fields and Cayley–Dickson algebras (the latter terms are absent in the associative case), see [1][3]. Let $ A $ be a two-sided ideal of the ring $ R $; then

$$ J ( A) = A \cap J ( R) $$

(see [1], [4]); if $ R $ is an associative ring and $ R _ {n} $ is the ring of matrices of order $ n $ over $ R $, then

$$ J ( R _ {n} ) = [ J ( R) ] _ {n} . $$

If $ R $ is an associative algebra over a field $ F $ and the cardinality of $ F $ is greater than the dimension of $ R $ over $ F $ or if $ R $ is algebraic over $ F $, then $ J ( R) $ is a nil ideal. A quasi-regular radical of a finitely-generated alternative ring satisfying an essential identity relation is the same as a lower nil radical (see Radical of rings and algebras) [6]. Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. Jordan algebra).

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2] K.A. Zhevlakov, "On radical ideals of an alternative ring" Algebra i Logika , 4 : 4 (1965) pp. 87–102 (In Russian)
[3] K.A. Zhevlakov, "Alternative Artinian rings" Algebra i Logika , 5 : 3 (1966) pp. 11–36 (In Russian)
[4] K.A. Zhevlakov, "On the Kleinfeld and Smiley radicals of alternative rings" Algebra and Logic , 8 : 2 (1969) pp. 100–102 Algebra i Logika , 8 : 2 (1969) pp. 176–180
[5] K.A. Zhevlakov, "Coincidence of Kleinfeld and Smiley radicals in alternative rings" Algebra and Logic , 8 : 3 (1969) pp. 175–181 Algebra i Logika , 8 : 3 (1969) pp. 309–319
[6] K.A. Zhevlakov, "Quasiregular ideals in finitely generated alternative rings" Algebra and Logic , 11 : 2 (1972) pp. 94–100 Algebra i Logika , 11 : 2 (1972) pp. 140–1161
[7] A.M. Slin'ko, I.P. Shestakov, "Right representation of algebras" Algebra and Logic , 13 : 5 (1973) pp. 312–333 Algebra i Logika , 13 : 5 (1974) pp. 544–588
[8] E. Kleinfeld, "Primitive alternative rings and semi-simplicity" Amer. J. Math. , 77 (1955) pp. 725–730
[9] K. McCrimmon, "The radical of a Jordan algebra" Proc. Nat. Acad. Sci. USA , 62 (1969) pp. 671–678
[10] M.F. Smiley, "The radical of an alternative ring" Ann. of Math. , 49 : 3 (1948) pp. 702–709

Comments

The radical of a row-finite infinite matrix ring is determined by annihilating sequences of ideals [a1]. The radical of a polynomial ring $ R [ x] $ is $ N [ x] $ for some nil ideal $ N $ in $ R $( the determination of $ N $ is an open problem).

References

[a1] N.E. Sexauer, J.E. Warnock, "The radical of the row-finite matrices over an arbitrary ring" Trans. Amer. Math. Soc. , 39 (1969) pp. 281–295
[a2] L.H. Rowen, "Ring theory" , I, II , Acad. Press (1988)
How to Cite This Entry:
Quasi-regular radical. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-regular_radical&oldid=48393
This article was adapted from an original article by I.P. Shestakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article