Jacobson radical

In mathematics, more specifically ring theory, the Jacobson radical of a ring formula_1 is the ideal consisting of those elements in formula_1 that annihilate all simple right formula_1-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 formula_4 or formula_5; 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 .
The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. 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.
Definitions.
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 formula_1 is defined as the intersection of all maximal ideals formula_7. If we denoted formula_8 as the set of all maximal ideals in formula_1 thenformula_10This definition can be used for explicit calculations in a number of simple cases, such as for local rings formula_11, which have a unique maximal ideal, Artin rings, and products thereof. See the examples section for explicit computations.
Noncommutative/general case.
For a general ring with unit formula_1, the Jacobson radical formula_4 is defined as the ideal of all elements formula_14 such that formula_15 whenever formula_16 is a simple module. That is,formula_17This is equivalent to the definition in the commutative case for a commutative ring formula_1 because the simple modules over a commutative ring are of the form formula_19 for some maximal ideal formula_20, and the only annihilators of formula_19 in formula_1 are in formula_7, i.e. formula_24.
Motivation.
Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations.
Geometric applications.
Although Jacobson originally introduced his radical as a technique for build 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 Nakayama's lemma. It is a technical tool for studying finitely generated modules over commutative rings which has an easy geometric interpretation: If we have a vector bundle formula_25 over a topological space formula_26, and pick a point formula_27, then any basis of formula_28 can be extended to a basis of sections of formula_29 for some neighborhood formula_30.
Another application is in the case of finitely generated commutative rings, meaning formula_1 is of the formformula_32for some base ring formula_33 (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 formula_34 defining the ring formula_1 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 which 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 , , and .
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 for "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 :