View source for Quasi-regular radical

''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 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

<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>

(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

<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>

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]]).

====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 <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).

====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>