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The Jacobson radical

The Jacobson radical typeset structure of a ring typeset structure with identity is defined by one of the following equivalent ways:

  • typeset structure the intersection of all maximal left ideals
  • typeset structure the intersection of all maximal left ideals
  • consequently if typeset structure is commutative typeset structure is the intersection of all maximal ideals in typeset structure. Note that if typeset structure is not commutative, then typeset structure is not necessarily equal to the intersection of all two-sided maximal ideals in typeset structure.
  • typeset structure is the largest ideal typeset structure such that for all typeset structure the element typeset structure is invertible in typeset structure (the converse statement, that if typeset structure is such that typeset structure is invertible then typeset structure is not true).
  • typeset structure
  • typeset structure
  • The Jacobson radical is named for Nathan Jacobson , who first defined and studied the Jacobson radical.

    If ring does not have identity, then the definition of the Jacobson ideal is more involved.  First define the binary operation typeset structure on typeset structure by typeset structure for all typeset structure. Then typeset structure is a semigroup with an identity element. Then the Jacobson radical is defined to be the largest ideal typeset structure of typeset structure such that typeset structure is a subgroup of typeset structure.

    Cite this web-page as:

    Štefan Porubský: The Jacobson radical.

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