View source for Quasi-regular radical

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''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|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  $  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

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

(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

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

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|Radical of rings and algebras]]) [[#References|[6]]]. Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. [[Jordan algebra|Jordan algebra]]).

====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>

====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  $  R [ x] $
is  $  N [ x] $
for some nil ideal  $  N $
in  $  R $(
the determination of  $  N $
is an open problem).

====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>