Jacobson radical

In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules.

It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric.

The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring.

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to non-unital rings.

The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as Nakayama's lemma.

This definition can be used for explicit calculations in a number of simple cases, such as for local rings (R,

), which have a unique maximal ideal, Artinian rings, and products thereof.

For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.

Although Jacobson originally introduced his radical as a technique for building a theory of radicals for arbitrary rings, one of the motivating reasons for why the Jacobson radical is considered in the commutative case is because of its appearance in Nakayama's lemma.

This lemma is a technical tool for studying finitely generated modules over commutative rings that has an easy geometric interpretation: If we have a vector bundle E → X over a topological space X, and pick a point p ∈ X, then any basis of E|p can be extended to a basis of sections of E|U → U for some neighborhood p ∈ U ⊆ X.

Another application is in the case of finitely generated commutative rings of the form

This means we could interpret the Jacobson radical as a measure for how far the ideal I defining the ring R is from defining the ring of functions on an algebraic variety because of the Hilbert Nullstellensatz theorem.

This is because algebraic varieties cannot have a ring of functions with infinitesimals: this is a structure that is only considered in scheme theory.

The Jacobson radical of a ring has various internal and external characterizations.

The following equivalences appear in many noncommutative algebra texts such as Anderson & Fuller 1992, §15, Isaacs 1994, §13B, and Lam 2001, Ch 2.