Encyclopedia > Jacobson radical

  Article Content

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:

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

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article

... regions. Its moisture content usually is less then 20 percent. The heat content of bituminous coal ranges from 21 to 30 million Btu per ton on a moist, ...