Nil ideal
From Encyclopedia of Mathematics
A subset 
of a ring 
is called nil if each element of it is nilpotent (cf. Nilpotent element). An ideal of 
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
 |
where 
denotes the Jacobson radical of 
and 
is the prime radical of 
, i.e. the intersection of all prime ideals of 
. Each of the inclusions can be proper. If 
is commutative, 
. The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.
References
| [a1] | C. Faith, "Algebra" , II. Ring theory , Springer (1976) |
| [a2] | J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) |
| [a3] | L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) |
How to Cite This Entry:
Nil ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_ideal&oldid=12426
Nil ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_ideal&oldid=12426