Jacobson radical

In ring theory, a branch of 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".

Definition

The Jacobson radical is denoted by J("R") and can be defined in the following equivalent ways:
* the intersection of all maximal left ideals.
* the intersection of all maximal right ideals.
* the intersection of all annihilators of simple left "R"-modules
* the intersection of all annihilators of simple right "R"-modules
* the intersection of all left primitive ideals.
* the intersection of all right primitive ideals.
* { "x" ∈ "R" : for every "r" ∈ "R" there exists "u" ∈ "R" with "u" (1-"rx") = 1 }
* { "x" ∈ "R" : for every "r" ∈ "R" there exists "u" ∈ "R" with (1-"xr") "u" = 1 }
* if "R" is commutative, the intersection of all maximal ideals in "R".
* the largest ideal "I" such that for all "x" ∈ "I", 1-"x" is invertible in "R"

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".

A Jacobson radical may also be defined for rings without an identity (or unity) element. See Noncommutative Rings by I. N. Herstein

The Jacobson radical is named for Nathan Jacobson, who first studied the Jacobson radical.

Examples

* 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.
* If "K" is a field and "R" = "K""X"₁,...,"X"_"n" is a ring of formal power series, then J("R") consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring consists precisely of the ring's non-units.
* Start with a finite quiver Γ and a field "K" and consider the quiver algebra "K"Γ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
* The Jacobson radical of a C*-algebra is {0}. This follows from the Gelfand–Naimark theorem and the fact for a C*-algebra, a topologically irreducible *-representation on a Hilbert space is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (see spectrum of a C*-algebra).

Properties

* Unless "R" is the trivial ring {0}, the Jacobson radical is always an ideal in "R" distinct from "R".

* If "R" is commutative and finitely generated as a Z-module, 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 rings.

* 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 consist of only the nilpotent elements of the ring.

* "R" is a semisimple ring if and only if it is Artinian and its Jacobson radical is zero.

ee also

*Radical of a module
*Radical of an ideal

References

*M. F. Atiyah, I. G. Macdonald. "Introduction to Commutative Algebra".
*N. Bourbaki. "Éléments de Mathématique".
*I. N. Herstein, "Noncommutative Rings".
*R. S. Pierce. "Associative Algebras". Graduate Texts in Mathematics vol 88.
*T. Y. Lam. "A First Course in Non-commutative Rings". Graduate Texts in Mathematics vol 131.

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