Editing Jacobson radical

{{Short description|Structure in Ring Theory (Mathematics)}}
[[File:Nathan_Jacobson.jpg | thumb|220x124px | right | alt= Image of Nathan Jacobson looking off-camera in a sitting position  | Nathan Jacobson]]In [[mathematics]], more specifically [[ring theory]], the '''Jacobson radical''' of a [[Ring (mathematics)|ring]] ''R'' is the [[ideal (ring theory)|ideal]] consisting of those elements in ''R'' that [[annihilator (ring theory)|annihilate]] all [[Simple module|simple]] right ''R''-[[module (mathematics)|modules]]. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric. The Jacobson radical of a ring is frequently denoted by J(''R'') or rad(''R''); the former notation will be preferred in this article, because it avoids confusion with other [[radical of a ring|radicals of a ring]]. The Jacobson radical is named after [[Nathan Jacobson]], who was the first to study it for arbitrary rings in {{harvnb|Jacobson|1945}}.

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to [[non-unital ring]]s. The [[radical of a module]] extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as [[Nakayama's lemma]].
<!-- For instance, if ''R'' is a ring, J(''R'') equals the intersection of all ''maximal right ideals'' in ''R''.{{sfn|ps=|Isaacs|1994|p=180|loc=Corollary 13.3}} Somewhat remarkable is that this also equals the intersection of all ''maximal left ideals'' of ''R''.{{sfn|ps=|Isaacs|1994|p=182}} Although the Jacobson radical is indeed an ideal, this is not entirely obvious from the previous two characterizations and hence other characterizations are preferred.{{sfn|ps=|Isaacs|1994|p=180}} Despite the nature of these characterizations, the intersection of all ''maximal (double-sided) ideals'' in ''R'' need not equal J(''R'') – for instance, when ''R'' is a the [[endomorphism ring]] of a [[vector space]] with [[countable]] [[dimension of a vector space|dimension]] over a field ''F'', it is known that ''R'' has precisely three ideals, {{mset|0}}, ''I'' and ''R'', however since ''R'' is [[von Neumann regular]] {{nowrap|1=J(''R'') = 0++}}. {{harv|Lam|2001|p=46|loc=Ex. 3.15}}-->
<!-- A computationally convenient notion when working with the Jacobson radical of a ring, is the notion of [[Quasiregular element|quasiregularity]].{{sfn|ps=|Isaacs|1994|p=180}} In particular, every element of a ring's Jacobson radical is quasiregular, and the Jacobson radical can be characterized as the unique right ideal of a ring, maximal with respect to the property that each element is [[Quasiregular element|right quasiregular]].{{sfn|ps=|Isaacs|1994|p=180|loc=Theorem 13.4}}{{sfn|ps=|Isaacs|1994|p=181}} It is not necessarily true, however, that every quasiregular element belongs to a ring's Jacobson radical.{{sfn|ps=|Isaacs|1994|p=181}} The notion of quasiregularity proves to be very useful in various situations discussed later{{sfn|ps=|Isaacs|1994|p=181}}{{sfn|ps=|Isaacs|1994|p=183|loc=Theorem 13.11}} -->
<!--The Jacobson radical of a ring is also useful in studying [[Module (mathematics)|modules]] over the ring.{{sfn|ps=|Isaacs|1994|p=182}}{{sfn|ps=|Isaacs|1994|p=183|loc=Theorem 13.11}} For instance, if ''U'' is a right ''R''-module, and ''V'' is a maximal submodule of ''U'', then {{nowrap|''U'' · J(''R'')}} is contained in ''V'', where {{nowrap|''U'' · J(''R'')}} denotes all products of elements of J(''R'') (the "scalars") with elements in ''U'', on the right.{{sfn|ps=|Isaacs|1994|p=182}} Another instance of the usefulness of J(''R'') when studying right ''R''-modules, is [[Nakayama's lemma]].{{sfn|ps=|Isaacs|1994|p=183|loc=Corollary 13.12}}-->
<!-- In this case, the ring may not even contain a (proper) ''maximal'' right or left ideal (although, it may well contain non-trivial proper (one-sided) ideals). Thus, all of the above characterizations fail (including the characterization involving [[Quasiregular element|quasiregularity]] for this requires that the ring have unity). This problem, as well as the solution, is discussed later in the article, where the Jacobson radical is defined for rings without unity. -->

== Definitions ==
There are multiple equivalent definitions and characterizations of the Jacobson radical, but it is useful to consider the definitions based on if the ring is [[commutative ring|commutative]] or not.

=== Commutative case ===
In the commutative case, the Jacobson radical of a commutative ring ''R'' is defined as<ref>{{Cite web|title=Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project|url=https://stacks.math.columbia.edu/tag/0AMD|access-date=2020-12-24|website=stacks.math.columbia.edu}}</ref> the [[intersection (set theory)|intersection]] of all [[maximal ideal]]s <math>\mathfrak{m}</math>. If we denote {{nowrap|Specm ''R''}} as the set of all maximal ideals in ''R'' then<blockquote><math>\mathrm{J}(R) = \bigcap_{
    \mathfrak{m} \,\in\, \operatorname{Specm}R
} \mathfrak{m}</math></blockquote>This definition can be used for explicit calculations in a number of simple cases, such as for [[local ring]]s {{nowrap|(''R'', <math>\mathfrak{p}</math>)}}, which have a unique maximal ideal, [[Artinian ring]]s, and [[product ring|products]] thereof. See the examples section for explicit computations.

=== Noncommutative/general case ===
For a general ring with unity ''R'', the Jacobson radical J(''R'') is defined as the ideal of all elements {{nowrap|''r'' ∈ ''R''}} such that {{nowrap|1=''rM'' = 0}} whenever ''M'' is a [[simple module|simple]] ''R''-module. That is,
<math display="block">\mathrm{J}(R) = \{r \in R \mid rM = 0 \text{ for all } M \text{ simple} \}.</math>
This is equivalent to the definition in the commutative case for a commutative ring ''R'' because the simple modules over a commutative ring are of the form {{nowrap|''R'' / <math>\mathfrak{m}</math>}} for some maximal ideal {{nowrap|<math>\mathfrak{m}</math> of ''R''}}, and the [[annihilator (ring theory)|annihilators]] of {{nowrap|''R'' / <math>\mathfrak{m}</math>}} in ''R'' are precisely the elements of <math>\mathfrak{m}</math>, i.e. {{nowrap|1=Ann<sub>''R''</sub>(''R'' / <math>\mathfrak{m}</math>) = <math>\mathfrak{m}</math>}}.

== Motivation ==
Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting [[geometry|geometric]] interpretations, and its algebraic interpretations.

=== Geometric applications ===
{{see also|Nakayama's lemma#Geometric interpretation}}
Although Jacobson originally introduced his radical as a technique for building a theory of radicals for arbitrary rings, one of the motivating reasons for why the Jacobson radical is considered in the commutative case is because of its appearance in [[Nakayama's lemma]]. This lemma is a technical tool for studying [[finitely generated module]]s over commutative rings that has an easy geometric interpretation: If we have a [[vector bundle]] {{nowrap|''E'' → ''X''}} over a [[topological space]] ''X'', and pick a point {{nowrap|''p'' ∈ ''X''}}, then any basis of ''E''|<sub>''p''</sub> can be extended to a basis of sections of {{nowrap|''E''{{!}}<sub>''U''</sub> → ''U''}} for some [[neighborhood (topology)|neighborhood]] {{nowrap|''p'' ∈ ''U'' ⊆ ''X''}}.

Another application is in the case of finitely generated commutative rings of the form <math display="inline">R = k[x_1,\ldots, x_n]\,/\,I </math> for some base ring ''k'' (such as a [[field (mathematics)|field]], or the ring of [[Integer#Algebraic properties|integers]]). In this case the [[Nilradical of a ring|nilradical]] and the Jacobson radical coincide. This means we could interpret the Jacobson radical as a measure for how far the ideal ''I'' defining the ring ''R'' is from defining the ring of functions on an [[algebraic variety]] because of the [[Hilbert Nullstellensatz]] theorem. This is because algebraic varieties cannot have a ring of functions with infinitesimals: this is a structure that is only considered in [[Scheme (mathematics)|scheme theory]].

=== Equivalent characterizations ===
The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many [[noncommutative algebra]] texts such as {{harvnb|Anderson|Fuller|1992|loc=§15}}, {{harvnb|Isaacs|1994|loc=§13B}}, and {{harvnb|Lam|2001|loc=Ch 2}}.

The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward):

* J(''R'') equals the intersection of all [[maximal ideal|maximal right ideals]] of the ring. The equivalence coming from the fact that for all maximal right ideals ''M'', {{nowrap|''R'' / ''M''}} is a simple right ''R''-module, and that in fact all simple right ''R''-modules are [[isomorphic]] to one of this type via the map from ''R'' to ''S'' given by {{nowrap|''r'' ↦ ''xr''}} for any generator ''x'' of ''S''. It is also true that J(''R'') equals the intersection of all maximal left ideals within the ring.{{sfn|ps=|Isaacs|1994|p=182}} These characterizations are internal to the ring, since one only needs to find the maximal right ideals of the ring. For example, if a ring is [[Local ring|local]], and has a unique maximal ''right ideal'', then this unique maximal right ideal is exactly J(''R'').  Maximal ideals are in a sense easier to look for than annihilators of modules. This characterization is deficient, however, because it does not prove useful when working computationally with J(''R''). The left-right symmetry of these two definitions is remarkable and has various interesting consequences.{{sfn|ps=|Isaacs|1994|p=182}}{{sfn|ps=|Isaacs|1994|p=173|loc=Problem 12.5}} This symmetry stands in contrast to the lack of symmetry in the [[Socle of a ring|socles]] of ''R'', for it may happen that soc(''R''<sub>''R''</sub>) is not equal to soc(<sub>''R''</sub>''R''). If ''R'' is a [[non-commutative ring]], J(''R'') is not necessarily equal to the intersection of all maximal ''two-sided'' ideals of ''R''. For instance, if ''V'' is a [[countable set|countable]] direct sum of copies of a field ''k'' and {{nowrap|1=''R'' = End(''V'')}} (the [[ring of endomorphisms]] of ''V'' as a ''k''-module), then {{nowrap|1=J(''R'') = 0}} because ''R'' is known to be [[von Neumann regular]], but there is exactly one maximal double-sided ideal in ''R'' consisting of endomorphisms with finite-dimensional [[image (mathematics)|image]].{{sfn|ps=|Lam|2001|p=46|loc=Ex. 3.15}}
* J(''R'') equals the sum of all [[superfluous submodule|superfluous right ideals]] (or symmetrically, the sum of all superfluous left ideals) of ''R''. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection of maximal right ideals.  This phenomenon is reflected dually for the right socle of ''R''; soc(''R''<sub>''R''</sub>) is both the sum of [[minimal ideal|minimal right ideal]]s and the intersection of [[essential extension|essential right ideals]].  In fact, these two relationships hold for the radicals and socles of modules in general.
* As defined in the introduction, J(''R'') equals the intersection of all [[Annihilator (ring theory)|annihilators]] of [[simple module|simple]] right ''R''-modules, however it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilator of a simple module is known as a [[primitive ideal]], and so a reformulation of this states that the Jacobson radical is the intersection of all primitive ideals.  This characterization is useful when studying modules over rings. For instance, if ''U'' is a right ''R''-module, and ''V'' is a [[maximal submodule]] of ''U'', {{nowrap|''U'' · J(''R'')}} is contained in ''V'', where {{nowrap|''U'' · J(''R'')}} denotes all products of elements of J(''R'') (the "scalars") with elements in ''U'', on the right. This follows from the fact that the [[quotient module]] {{nowrap|''U'' / ''V''}} is simple and hence annihilated by J(''R'').
* J(''R'') is the unique right ideal of ''R'' maximal with the property that every element is [[Quasiregular element|right quasiregular]]{{sfn|ps=|Isaacs|1994|p=180|loc=Corollary 13.4}}{{sfn|ps=|Isaacs|1994|p=181}} (or equivalently left quasiregular{{sfn|ps=|Isaacs|1994|p=182}}). This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization is useful in studying modules over a ring. [[Nakayama's lemma]] is perhaps the most well-known instance of this. Although every element of the J(''R'') is necessarily [[Quasiregular element|quasiregular]], not every quasiregular element is necessarily a member of J(''R'').{{sfn|ps=|Isaacs|1994|p=181}}
* While not every quasiregular element is in J(''R''), it can be shown that ''y'' is in J(''R'') [[if and only if]] ''xy'' is left quasiregular for all ''x'' in ''R''.{{sfn|sp=|Lam|2001|p=50}}
* J(''R'') is the set of elements ''x'' in ''R'' such that every element of {{nowrap|1 + ''RxR''}} is a unit: {{nowrap|1=J(''R'') = {{mset|''x'' ∈ ''R'' {{pipe}} 1 + ''RxR'' ⊂ ''R''<sup>×</sup>}}}}. In fact, {{nowrap|''y'' ∈ ''R''}} is in the Jacobson radical if and only if {{nowrap|1 + ''xy''}} is invertible for any {{nowrap|''x'' ∈ ''R''}}, if and only if {{nowrap|1 + ''yx''}} is invertible for any {{nowrap|''x'' ∈ ''R''}}. This means ''xy'' and ''yx'' behave similarly to a [[nilpotent]] element ''z'' with {{nowrap|1={{italics correction|''z''}}<sup>''n''+1</sup> = 0}} and {{nowrap|1=(1 + ''z'')<sup>−1</sup> = 1 − ''z'' + {{italics correction|''z''}}<sup>2</sup> − ... ± {{italics correction|''z''}}<sup>''n''</sup>}}.

For [[Rng (algebra)|rings without unity]] it is possible to have {{nowrap|1=''R'' = J(''R'')}}; however, the equation {{nowrap|1=J(''R'' / J(''R'')) = {{mset|0}}}} still holds.  The following are equivalent characterizations of J(''R'') for rings without unity:{{sfn|ps=|Lam|2001|p=63}}
* The notion of left quasiregularity can be generalized in the following way.  Call an element ''a'' in ''R'' left ''generalized quasiregular'' if there exists ''c'' in ''R'' such that {{nowrap|1=''c'' + ''a'' − ''ca'' = 0}}.  Then J(''R'') consists of every element ''a'' for which ''ra'' is left generalized quasiregular for all ''r'' in ''R''.  It can be checked that this definition coincides with the previous quasiregular definition for rings with unity.
* For a ring without unity, the definition of a left simple module ''M'' is amended by adding the condition that {{nowrap|''R'' ⋅ ''M'' ≠ 0}}.  With this understanding, J(''R'') may be defined as the intersection of all annihilators of simple left ''R'' modules, or just ''R'' if there are no simple left ''R'' modules.  Rings without unity with no simple modules do exist, in which case {{nowrap|1=''R'' = J(''R'')}}, and the ring is called a '''radical ring'''.  By using the generalized quasiregular characterization of the radical, it is clear that if one finds a ring with J(''R'') nonzero, then J(''R'') is a radical ring when considered as a ring without unity.

== 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]]'').

== 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.

== See also ==
* [[Frattini subgroup]]
* [[Nilradical of a ring]]
* [[Radical of a module]]
* [[Radical of an ideal]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

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   | publisher = [[European Mathematical Society]]
   | title = International Congress of Mathematicians, Vol. II
   | url = http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_12.pdf
   | year = 2006
   | access-date = 2014-12-31
   | archive-date = 2017-08-09
   | archive-url = https://web.archive.org/web/20170809170625/http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_12.pdf
   | url-status = dead
   }}

== External links ==
* [https://mathoverflow.net/questions/3483/intuitive-example-of-a-jacobson-radical Intuitive Example of a Jacobson Radical]

{{DEFAULTSORT:Jacobson Radical}}
[[Category:Ideals (ring theory)]]
[[Category:Ring theory]]