View source for Nil ideal

<!--
n0666701.png
$#A+1 = 11 n = 0
$#C+1 = 11 : ~/encyclopedia/old_files/data/N066/N.0606670 Nil ideal
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A subset  $  A $
of a ring  $  R $
is called nil if each element of it is nilpotent (cf. [[Nilpotent element|Nilpotent element]]). An ideal of  $  R $
is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that

$$ 
 \mathop{\rm Jac} ( R)  \supset  \textrm{ Nil  Rad  } ( R)  \supset  \textrm{ Prime  Rad  } ( R),
$$

where  $   \mathop{\rm Jac} ( R) $
denotes the [[Jacobson radical|Jacobson radical]] of  $  R $
and  $  \textrm{ Prime  Rad  } ( R) $
is the prime radical of  $  R $, 
i.e. the intersection of all prime ideals of  $  R $. 
Each of the inclusions can be proper. If  $  R $
is commutative,  $  \textrm{ Nil  Rad  } ( R) = \textrm{ Prime  Rad  } ( R) $. 
The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,   "Algebra" , '''II. Ring theory''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,   J.C. Robson,   "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Rowen,   "Ring theory" , '''1''' , Acad. Press  (1988)</TD></TR></table>