Editing Jacobson radical (section)

== 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.