enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Radical of a ring - Wikipedia

    en.wikipedia.org/wiki/Radical_of_a_ring

    In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson ...

  3. Radical of an ideal - Wikipedia

    en.wikipedia.org/wiki/Radical_of_an_ideal

    Consider the ring of integers.. The radical of the ideal of integer multiples of is (the evens).; The radical of is .; The radical of is .; In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer).

  4. Jacobson radical - Wikipedia

    en.wikipedia.org/wiki/Jacobson_radical

    For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...

  5. Nil ideal - Wikipedia

    en.wikipedia.org/wiki/Nil_ideal

    Therefore, the set of all nilpotent elements forms an ideal known as the nil radical of a ring. Because the nil radical contains every nilpotent element, an ideal of a commutative ring is nil if and only if it is a subset of the nil radical, and so the nil radical is maximal among nil ideals. Furthermore, for any nilpotent element a of a ...

  6. Ideal (ring theory) - Wikipedia

    en.wikipedia.org/wiki/Ideal_(ring_theory)

    The factor ring of a radical ideal is a semiprime ring for general rings, ... Many classic examples of this stem from algebraic number theory. For example, ...

  7. Maximal ideal - Wikipedia

    en.wikipedia.org/wiki/Maximal_ideal

    If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R). It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example ...

  8. Ring theory - Wikipedia

    en.wikipedia.org/wiki/Ring_theory

    The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by ...

  9. Ideal theory - Wikipedia

    en.wikipedia.org/wiki/Ideal_theory

    In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a Noetherian integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's Algèbre commutative gives such a theory.