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:
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.
- 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?
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) from PlanetMath (http://www.planetmath.org).
All Wikipedia text
is available under the
terms of the GNU Free Documentation License