Jacobson radical

In ring theory, a branch of algebra, called the Jacobson radical of a ring R is an ideal of R that contains elements of R, which may be regarded as " close to zero ". The Jacobson radical is named after Nathan Jacobson, who has studied first.

  • 2.1 Definition
  • 2.2 Features
  • 2.3 Examples

Jacobson radical of R-modules

In the following, let R be a ring with unity and M is a left R- module.

Definition

The intersection of all maximal R- submodules of M is called the ( Jacobson ) radical ( or short). If M has no maximal submodules, so shall you.

If M is finitely generated, then:. This is called an element x of M superfluous if for each sub- module: For reasons already follows.

Properties

  • If M is finitely generated and a submodule of M, then it already is. This property is also referred to as a lemma Nakayama.
  • If M is finitely generated and, then. (This is the special case of the previous statement. )
  • Holds if and only if M is isomorphic simple R-modules is a submodule of a direct product.
  • M is then finitely generated and semisimple, if M is artinian and.

Jacobson radical of rings

In the following, let R be a ring with 1

Definition

Jacobson the radical R of the ring is defined as the radical R- Jacobson links R module. It is noted as J (R) and characterized by the following equivalent conditions:

  • As intersection of all maximal left ideals / right ideals
  • Easier than average for all Annullatoren left R-modules / right R-modules

Properties

  • The ring R is semisimple if and only if it is linksartinsch and J ( R) = 0.
  • For each linksartinschen ring R, the ring is semisimple.
  • If R linksartinsch, then for every left R- module M:.
  • J (R ) is the smallest ideal I of R such that R / I is semisimple.
  • N is a Nillinksideal of R, then:.
  • If R linksartinsch, then J (R ) is a nilpotent ideal.
  • If R linksartinsch, then the Jacobson radical is equal to the Primradikal.
  • By Zorn's lemma, for each ring the existence of maximal ideals, so for true.

Examples

  • Is the Jacobson radical of a skew field; as is the Jacobson radical of.
  • The Jacobson radical of is.
  • The Jacobson radical of the ring of all upper triangular matrices over a field K, contains those upper triangular matrices whose diagonal entries vanish.
  • The Jacobson radical of each local ring is its maximal ideal, that is just out of his non- units.
  • The Jacobson radical of a commutative Banach algebra is exactly the core of the Gelfand transform.
Nakayama lemma Semisimple module Artinian ring Simple module Nilideal Radical of an ideal Michiel Hazewinkel Encyclopedia of Mathematics
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