View source for Jacobson radical

''of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541701.png" />''

The [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541702.png" /> of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541703.png" /> which satisfies the following two requirements: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541704.png" /> is the largest quasi-regular ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541705.png" /> (a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541706.png" /> is called quasi-regular if the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541707.png" /> is solvable for any of its elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541708.png" />); and 2) the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j0541709.png" /> contains no non-zero quasi-regular ideals. The radical was introduced and studied in detail in 1945 by N. Jacobson [[#References|[1]]].

The Jacobson radical always exists and may be characterized in very many ways: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417010.png" /> is the intersection of the kernels of all irreducible representations of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417011.png" />; it is the intersection of all modular maximal right ideals (cf. [[Modular ideal|Modular ideal]]); it is the intersection of all modular maximal left ideals; it contains all quasi-regular one-sided ideals; it contains all one-sided nil ideals; etc. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417012.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417015.png" /> is the ring of all matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417017.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/j/j054/j054170/j05417018.png" /></td> </tr></table>

If the following <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417019.png" />-composition is introduced on the associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417020.png" />:

<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/j/j054/j054170/j05417021.png" /></td> </tr></table>

then the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417022.png" /> in the [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417023.png" /> will be a subgroup with respect to the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417024.png" />.

There are no non-zero irreducible finitely-generated modules over a quasi-regular associative ring (i.e. an associative ring coinciding with its own Jacobson radical), but there exist simple associative quasi-regular rings. The Jacobson radical of the associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417025.png" /> is zero if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417026.png" /> is a subdirect sum of primitive rings (cf. [[Primitive ring|Primitive ring]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>



====Comments====
The Jacobson radical is the intersection of the right primitive ideals. It is also the intersection of the left primitive ideals. This is perhaps the most frequently occurring definition. Modular ideals are also called regular ideals. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417027.png" /> has a unit element, then all ideals are regular, so that in this case the Jacobson radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417028.png" /> is the intersection of all right maximal ideals and also the intersection of all left maximal ideals. Nakayama's lemma says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417029.png" /> is a finitely-generated non-zero right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417030.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054170/j05417031.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. McConnell,   J.C. Robson,   "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>