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 is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Jacobson 1945.

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to non-unital rings. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as Nakayama's lemma.


There are multiple equivalent definitions and characterizations of the Jacobson radical, but it is useful to consider the definitions based on if the ring is commutative or not.

Commutative case

In the commutative case, the Jacobson radical of a commutative ring R is defined as[1] the intersection of all maximal ideals . If we denote Specm R as the set of all maximal ideals in R then

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. See the examples section for explicit computations.

Noncommutative/general case

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

This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are precisely the elements of , i.e. AnnR(R / ) = .


Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations.

Geometric applications

See also: Nakayama's lemma § Geometric interpretation

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 EX over a topological space X, and pick a point pX, then any basis of E|p can be extended to a basis of sections of E|UU for some neighborhood pUX.

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

for some base ring k (such as a field, or the ring of integers). In this case the nilradical and the Jacobson radical coincide. 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.

Equivalent characterizations

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.

The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward):

For rings without unity it is possible to have R = J(R); however, the equation J(R / J(R)) = {0} still holds. The following are equivalent characterizations of J(R) for rings without unity:[8]


Commutative examples

Noncommutative examples


See also


  1. ^ Proof: Since the factors Tu / Tu−1 are simple right R-modules, right multiplication by any element of J(R) annihilates these factors.
    In other words, (Tu / Tu−1) ⋅ J(R) = 0, whence Tu · J(R) ⊆ Tu−1. Consequently, induction over i shows that all nonnegative integers i and u (for which the following makes sense) satisfy Tu ⋅ (J(R))iTui. Applying this to u = i = k yields the result.


  1. ^ "Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project". Retrieved 2020-12-24.
  2. ^ a b c Isaacs 1994, p. 182
  3. ^ Isaacs 1994, p. 173, Problem 12.5
  4. ^ Lam 2001, p. 46, Ex. 3.15
  5. ^ Isaacs 1994, p. 180, Corollary 13.4
  6. ^ a b Isaacs 1994, p. 181
  7. ^ Lam 2001, p. 50.
  8. ^ Lam 2001, p. 63
  9. ^ Smoktunowicz 2006, p. 260, §5