Nil ideal

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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.


[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)
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