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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: 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.



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.