Jacobson radical
In
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 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.
- 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 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. Note however that in general the Jacobson radical need not contain every nilpotent element of the ring.
See also: radical of a module.
This article (or an earlier version of it) was based on the Jacobson radical article from PlanetMath.