- Jacobson radical
In
ring theory , a branch ofabstract 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 leftprimitive ideal s.
* 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
integer s is {0}.
* The Jacobson radical of the ring Z/8Z (seemodular 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 everylocal 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 aC*-algebra is {0}. This follows from theGelfand–Naimark theorem and the fact for a C*-algebra, a topologically irreducible *-representation on aHilbert space is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (seespectrum 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 ring s.* 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 anilpotent ideal . Note however that in general the Jacobson radical need not consist of only thenilpotent elements of the ring.* "R" is a
semisimple ringif and only if it isArtinian 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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