Nilpotent ideal

In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I^{Template:Hair spacek} = 0.Template:Sfn By I^{Template:Hair spacek}, it is meant the additive subgroup generated by the set of all products of k elements in I.Template:Sfn Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0.

The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem.Template:SfnTemplate:Sfn The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

Relation to nil ideals

The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than one reason. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.Template:Sfn

In a right Artinian ring, any nil ideal is nilpotent.Template:Sfn This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis), the result follows. In fact, this can be generalized to right Noetherian rings; this result is known as Levitzky's theorem.Template:Sfn

See also

• Köthe conjecture
• Nilpotent element
• Nilradical
• Jacobson radical