# Nil ideal

A subset $ A $
of a ring $ R $
is called nil if each element of it is nilpotent (cf. 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 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

[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=47973