Jacobson radical

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

Jacobson radical of R modules

The following is a ring with one and an R-left module .


The average of all maximum submodules of is called the (Jacobson) radical (or short ).

Is finitely generated, then: . An element of is called superfluous if the following applies to each sub-module : From already follows .


  • If it is finally generated and a sub-module of with , then it is already . This property is also known as the Nakayama lemma .
  • Is finally generated and then is . (This is the special case of the previous statement.)
  • if and only if isomorphic easier to a sub-module of a direct product is -modules.
  • is finitely generated and semi-simple if and only if artinian and is.

Jacobson radical of rings

Let the following be a ring with one.


The Jacobson radical of the ring is defined as the Jacobson radical of the left module . It is noted as and characterized by the following equivalent conditions:

  • as the average of all maximum left / right ideals
  • as the average of all cancellers of simple left - module / right- module


  • The ring is semi- simple if and only if it is leftartinian .
  • For any Linksartinian ring , the ring is semi-simple.
  • Is linksartinsch, then for all -Linksmodul : .
  • is the smallest ideal of having the property that is semi-simple.
  • Is a Nillinksideal by then: .
  • If leftartinian, then it is a nilpotent ideal.
  • If left-Artinian, then the Jacobson radical is the same as the prime radical .
  • With Zorn's lemma , the existence of maximal ideals follows for every ring , so for holds .


  • The Jacobson radical of an oblique body is ; likewise the Jacobson radical of .
  • The Jacobson radical of is .
  • The Jacobson radical of the ring of all upper triangular matrices over a body contains those upper triangular matrices whose diagonal entries vanish.
  • The Jacobson radical of every local ring is its maximum ideal, i.e. it consists precisely of its non-units.
  • The Jacobson radical of a commutative Banach algebra is exactly the core of the Gelfand transformation .