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".

Contents

Definition

Examples

Properties

See also

References

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''.
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'/8'Z' (see modular arithmetic) is 2'Z'/8'Z'.

★ 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, 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.

See 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''.

★ 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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