Jacobson radical

Jacobson radical

The Jacobson radical of a ring R is an ideal of R which in a sense contains "superfluous" elements of R which 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.

Examples:

The Jacobson radical of any field is {0}. The Jacobson radical of the integers is {0}.
The Jacobson radical of the ring Z/8Z (see modular arithmetic) is 2Z/8Z.
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.
Start with a finite quiver Γ and a field K and consider the quiver algebra KΓ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
some more examples of non-trivial Jacobson radicals would be nice. Rings of continuous functions? Endomorphism rings?

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[?].

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[?].

See also: radical of a module[?].

This article (or an earlier version of it) was based on the Jacobson radical article (http://www.planetmath.org/encyclopedia/JacobsonRadical.html) from PlanetMath (http://www.planetmath.org).