In mathematics, more specifically ring theory, an idealI of a ringR is said to be a nilpotent ideal if there exists a natural numberk such that Ik = 0.[1] By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I.[1] 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.[2][3] The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
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.[1]
In a right Artinian ring, any nil ideal is nilpotent.[4] 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.[3]
nilpotent, ideal, mathematics, more, specifically, ring, theory, ideal, ring, said, nilpotent, ideal, there, exists, natural, number, such, that, meant, additive, subgroup, generated, products, elements, therefore, nilpotent, only, there, natural, number, such. 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 k 0 1 By I k it is meant the additive subgroup generated by the set of all products of k elements in I 1 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 2 3 The notion of a nilpotent ideal although interesting in the case of commutative rings is most interesting in the case of noncommutative rings Contents 1 Relation to nil ideals 2 See also 3 Notes 4 ReferencesRelation to nil ideals editThe 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 1 In a right Artinian ring any nil ideal is nilpotent 4 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 3 See also editKothe conjecture Nilpotent element Nilradical Jacobson radicalNotes edit a b c Isaacs 1993 p 194 Isaacs 1993 Theorem 14 38 p 210 a b Herstein 1968 Theorem 1 4 5 p 37 Isaacs 1993 Corollary 14 3 p 195 References editHerstein I N 1968 Noncommutative rings 1st ed The Mathematical Association of America ISBN 0 88385 015 X Isaacs I Martin 1993 Algebra a graduate course 1st ed Brooks Cole Publishing Company ISBN 0 534 19002 2 Retrieved from https en wikipedia org w index php title Nilpotent ideal amp oldid 1173250589, wikipedia, wiki, book, books, library,
