Nilpotent ideal

Nilpotent ideal

In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k 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.[2] 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.[3][4] The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

Contents

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 reason than one. 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.[5]

In a right artinian ring, any nil ideal is nilpotent.[6] 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 may generalize to right noetherian rings; a phenomenon known as Levitzky's theorem, a particularly simple proof of which is due to Utumi.[7]

See also

Notes

  1. ^ Isaacs, p. 194
  2. ^ Isaacs, p. 194
  3. ^ Isaacs, Theorem 14.38, p. 210
  4. ^ Herstein, Theorem 1.4.5, p. 37
  5. ^ Isaacs, p. 194
  6. ^ Isaacs, Corollary 14.3, p. 195
  7. ^ Herstein, Theorem 1.4.5, p. 37

References

  • I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America. ISBN 0-88385-015-X. 
  • I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2. 

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • nilpotent —   [zu lateinisch nihil, nil »nichts«], Mathematik: 1) Ein Element a eines Ringes heißt nilpotent, wenn es eine natürliche Zahl m gibt, sodass am = 0 ist; z. B. ist   im Ring der reellen 2 ☓ 2 Matrizen nilpotent wegen a2 = 0.   2) Ein (Rechts oder …   Universal-Lexikon

  • Nilpotent — This article is about a type of element in a ring. For the type of group, see Nilpotent group. In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. The term was… …   Wikipedia

  • Ideal (Lie-Algebra) — Lie Algebra berührt die Spezialgebiete Mathematik Lineare Algebra Lie Gruppen Physik Eichtheorie ist Spezialfall von Vektorraum …   Deutsch Wikipedia

  • Nilpotent Lie algebra — In mathematics, a Lie algebra is nilpotent if the lower central series becomes zero eventually. Equivalently, is nilpotent if …   Wikipedia

  • Nilpotent —  Ne doit pas être confondu avec Groupe nilpotent. En mathématiques, un élément x d un anneau unitaire (ou même d un pseudo anneau) R est appelé nilpotent s il existe un certain nombre entier positif n tel que . Sommaire …   Wikipédia en Français

  • Idéal — Pour les articles homonymes, voir Idéal (homonymie). En mathématiques, et plus particulièrement en algèbre, un idéal est un sous ensemble remarquable d un anneau. Par certains égards, les idéaux s apparentent aux sous espaces vectoriels ce sont… …   Wikipédia en Français

  • Nil ideal — In mathematics, more specifically ring theory, an ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.[1][2] The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring… …   Wikipedia

  • Radical of an ideal — In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. There are several special radicals associated with the entire ring such as the nilradical and the Jacobson radical , which isolate certain bad… …   Wikipedia

  • Nilradical of a ring — For more radicals, see radical of a ring. In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring. In the non commutative ring case, more care is needed resulting in several related radicals …   Wikipedia

  • Glossary of ring theory — Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”