Editing Jacobson radical (section)

== Examples ==

=== Commutative examples ===
* For the ring of integers '''Z''' its Jacobson radical is the [[zero ideal]], so {{nowrap|1=J('''Z''') = (0)}}, because it is given by the intersection of every ideal generated by a [[prime number]] (''p''). Since {{nowrap|1=(''p''<sub>1</sub>) ∩ (''p''<sub>2</sub>) = (''p''<sub>1</sub> ⋅ ''p''<sub>2</sub>)}}, and we are taking an infinite intersection with no common elements besides 0 between all maximal ideals, we have the computation.
* For a [[local ring]] {{nowrap|(''R'', <math>\mathfrak{p}</math>)}} the Jacobson radical is simply {{nowrap|1=J(''R'') = <math>\mathfrak{p}</math>}}. This is an important case because of its use in applying Nakayama's lemma. In particular, it implies if we have an algebraic vector bundle {{nowrap|''E'' → ''X''}} over a scheme or algebraic variety ''X'', and we fix a basis of ''E''|<sub>''p''</sub> for some point {{nowrap|''p'' ∈ ''X''}}, then this basis lifts to a set of generators for all sections {{nowrap|''E''|<sub>''U''</sub> → ''U''}} for some neighborhood ''U'' of ''p''.
* If ''k'' is a [[field (mathematics)|field]] and {{nowrap|1=''R'' = ''k''[[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>]]}} is a ring of [[formal power series]], then J(''R'') consists of those [[power series]] whose constant term is zero, i.e. the power series in the ideal {{nowrap|(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}.
* In the case of an [[Artinian ring]]s, such as {{nowrap|'''C'''[''t''<sub>1</sub>, ''t''<sub>2</sub>]/(''t''<sub>1</sub><sup>4</sup>, ''t''<sub>1</sub><sup>2</sup>''t''<sub>2</sub><sup>2</sup>, ''t''<sub>2</sub><sup>9</sup>)}}, the Jacobson radical is {{nowrap|(''t''<sub>1</sub>, ''t''<sub>2</sub>)}}.
* The previous example could be extended to the ring {{nowrap|1=''R'' = '''C'''[''t''<sub>2</sub>, ''t''<sub>3</sub>, ...]/(''t''<sub>2</sub><sup>2</sup>, ''t''<sub>3</sub><sup>3</sup>, ...)}}, giving {{nowrap|1=J(''R'') = (''t''<sub>2</sub>, ''t''<sub>3</sub>, ...)}}.
* The Jacobson radical of the ring '''Z'''/12'''Z''' is 6'''Z'''/12'''Z''', which is the intersection of the maximal ideals 2'''Z'''/12'''Z''' and 3'''Z'''/12'''Z'''.
* Consider the ring {{nowrap|'''C'''[''t''] ⊗<sub>'''C'''</sub> '''C'''[''x''<sub>1</sub>, ''x''<sub>2</sub>]<sub>''x''<sub>1</sub><sup>2</sup>+''x''<sub>2</sub><sup>2</sup>−1</sub>}}, where the second is the [[localization (commutative algebra)|localization]] of {{nowrap|'''C'''[''x''<sub>1</sub>, ''x''<sub>2</sub>]}} by the prime ideal {{nowrap|1=<math>\mathfrak{p}</math> = (''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> − 1)}}. Then, the Jacobson radical is trivial because the maximal ideals are generated by an element of the form {{nowrap|(''t'' − ''z'') ⊗ (''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> − 1)}} for {{nowrap|''z'' ∈ '''C'''}}.

=== Noncommutative examples ===
* Rings for which J(''R'') is {{mset|0}} are called [[semiprimitive ring]]s, or sometimes "Jacobson semisimple rings".  The Jacobson radical of any field, any [[von Neumann regular ring]] and any left or right [[primitive ring]]  is {{mset|0}}. The Jacobson radical of the integers is {{mset|0}}.
* If ''K'' is a field and ''R'' is the ring of all [[upper triangular]] ''n''-by-''n'' [[matrix (mathematics)|matrices]] with entries in ''K'', then J(''R'') consists of all upper triangular matrices with zeros on the main diagonal.
* Start with a finite, acyclic [[Quiver (mathematics)|quiver]] Γ and a field ''K'' and consider the quiver algebra ''K''{{Hair space}}Γ (as described in the article ''[[Quiver (mathematics)|Quiver]]''). The Jacobson radical of this ring is generated by all the paths in Γ of length&nbsp;≥&nbsp;1.
*  The Jacobson radical of a [[C*-algebra]] is {{mset|0}}. This follows from the [[Gelfand–Naimark theorem]] and the fact that 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]]'').