# Jacobson radical

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 .

### definition

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 .

### properties

- 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.

### definition

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

### properties

- 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 .

### Examples

- 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 .

## literature

- KA Zhevlakov: Jacobson radical . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).