# Jacobson radical

In ring theory, a branch of abstract algebra, the **Jacobson radical** of a ring *R* is an ideal of *R* which contains those elements of *R* which in a sense are "close to zero".
It is denoted by J(*R*) and can be defined in the following equivalent ways:

- the intersection of all maximal left ideals.
- the intersection of all maximal right ideals.
- the intersection of all annihilators of simple left
*R*-modules - the intersection of all annihilators of simple right
*R*-modules - the intersection of all left primitive ideals.
- the intersection of all right primitive ideals.
- {
*x*∈*R*: for every*r*∈*R*there exists*u*∈*R*with*u*(1-*rx*) = 1 } - {
*x*∈*R*: for every*r*∈*R*there exists*u*∈*R*with (1-*xr*)*u*= 1 } - the largest ideal
*I*such that for all*x*∈*I*, 1-*x*is invertible in*R*

Note that the last property does *not* mean that every element *x* of *R* such that 1-*x* is invertible must be an element of J(*R*).
Also, if *R* is not commutative, then J(*R*) is *not* necessarily equal to the intersection of all two-sided maximal ideals in *R*.

The Jacobson radical is named for Nathan Jacobson, who first studied the Jacobson radical.

#### Examples:

- The Jacobson radical of any field is {0}. The Jacobson radical of the integers is {0}.
- The Jacobson radical of the ring
**Z**/8**Z**(see modular arithmetic) is 2**Z**/8**Z**. - If
*K*is a field and*R*is the ring of all upper triangular*n*-by-*n*matrices with entries in*K*, then J(*R*) consists of all upper triangular matrices with zeros on the main diagonal. - If
*K*is a field and*R*=*K*[[*X*_{1},...,*X*_{n}]] is a ring of formal power series, then J(*R*) consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring consists precisely of the ring's non-units. - Start with a finite quiver Γ and a field
*K*and consider the quiver algebra*K*Γ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1. - The Jacobson radical of a C*-algebra is {0}. This follows from the Gelfand-Naimark theorem and the fact for a C*-algebra, a topologically irreducible *-representation on a Hilbert space is algebraically irreducible, so that its kernel is primitive ideal in the purely algebraic sense (see spectrum of a C*-algebra).

#### Properties

Unless *R* is the trivial ring {0}, the Jacobson radical is always a proper ideal in *R*.

If *R* is commutative and finitely generated, then J(*R*) is equal to the nilradical of *R*.

The Jacobson radical of the ring *R*/J(*R*) is zero. Rings with zero Jacobson radical are called semiprimitive rings.

If *f* : *R* `->` *S* is a surjective ring homomorphism, then *f*(J(*R*)) ⊆ J(*S*).

If *M* is a finitely generated left *R*-module with J(*R*)*M* = *M*, then *M* = 0 (*Nakayama lemma*).

J(*R*) contains every nil ideal of *R*. If *R* is left or right artinian, then J(*R*) is a nilpotent ideal. Note however that in general the Jacobson radical need not contain every nilpotent element of the ring.

**See also:** radical of a module, Radical_of_an_ideal.

*This article (or an earlier version of it) was based on the Jacobson radical article from PlanetMath.*