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''.
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''>''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 contain every Nilpotent element 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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{{planetmath|id=2856|title=Jacobson radical}}
Category:Ideals
De:Jacobson-Radikal