Editing Jacobson radical (section)

== Properties ==
* If ''R'' is unital and is not the trivial ring {{mset|0}}, the Jacobson radical is always distinct from ''R'' since [[Maximal ideal#Properties|rings with unity always have maximal right ideals]]. However, some important [[theorem]]s and [[conjecture]]s in ring theory consider the case when {{nowrap|1=J(''R'') = ''R''}} – "If ''R'' is a nil ring (that is, each of its elements is [[nilpotent]]), is the [[polynomial ring]] ''R''[''x''] equal to its Jacobson radical?" is equivalent to the open [[Köthe conjecture]].{{sfn|ps=|Smoktunowicz|2006|p=260|loc=§5}}
* For any ideal ''I'' contained in J(''R''), {{nowrap|1=J(''R'' / ''I'') = J(''R'') / ''I''}}.
* In particular, the Jacobson radical of the ring {{nowrap|''R'' / J(''R'')}} is zero. Rings with zero Jacobson radical are called [[semiprimitive ring]]s. 
* A ring is [[Semisimple module#Semisimple rings|semisimple]] if and only if it is [[Artinian ring|Artinian]] and its Jacobson radical is zero.
* If {{nowrap|''f'' : ''R'' → ''S''}} is a [[surjective]] [[ring homomorphism]], then {{nowrap|''f''(J(''R'')) ⊆ J(''S'')}}.
* If ''R'' is a ring with unity and ''M'' is a [[finitely generated module|finitely generated]] left ''R''-module with {{nowrap|1=J(''R'')''M'' = ''M''}}, then {{nowrap|1=''M'' = 0}} ([[Nakayama's lemma]]).
* J(''R'') contains all [[center (ring theory)|central]] nilpotent elements, but contains no [[idempotent (ring theory)|idempotent elements]] except for 0.
* J(''R'') contains every [[nil ideal]] of ''R''. If ''R'' is left or right [[Artinian ring|Artinian]], then J(''R'') is a [[nilpotent ideal]].{{pb}}This can actually be made stronger: If {{br}}{{spaces|8}}{{nowrap|1={{mset|0}} = ''T''<sub>0</sub> ⊆ ''T''<sub>1</sub> ⊆ ⋅⋅⋅ ⊆ ''T''<sub>''k''</sub> = ''R''}} {{br}}is a [[Composition series#For modules|composition series]] for the right ''R''-module ''R'' (such a series is sure to exist if ''R'' is right Artinian, and there is a similar left composition series if ''R'' is left Artinian), then {{nowrap|1=(J(''R''))<sup>''k''</sup> = 0}}.{{efn|1=Proof: Since the [[factor module|factors]] {{nowrap|''T''<sub>''u''</sub> / ''T''<sub>''u''−1</sub>}} are simple right ''R''-modules, right multiplication by any element of J(''R'') annihilates these factors.
{{br}}In other words, {{nowrap|1=(''T''<sub>''u''</sub> / ''T''<sub>''u''−1</sub>) ⋅ J(''R'') = 0}}, whence {{nowrap|1=''T''<sub>''u''</sub> · J(''R'') ⊆ ''T''<sub>''u''−1</sub>}}. Consequently, [[mathematical induction|induction]] over ''i'' shows that all nonnegative integers ''i'' and ''u'' (for which the following makes sense) satisfy {{nowrap|''T''<sub>''u''</sub> ⋅ (J(''R''))<sup>''i''</sup> ⊆ ''T''<sub>''u''−''i''</sub>}}. Applying this to {{nowrap|1=''u'' = ''i'' = ''k''}} yields the result.}}{{pb}}Note, however, that in general the Jacobson radical need not consist of only the [[nilpotent]] elements of the ring.
* If ''R'' is commutative and finitely generated as an [[algebra over a field|algebra]] over either a [[field (mathematics)|field]] or '''Z''', then J(''R'') is equal to the [[nilradical of a ring|nilradical]] of ''R''.
* The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.