Jacobson radical

Inmathematics,more specificallyring theory,theJacobson radicalof aringRis theidealconsisting of those elements inRthatannihilateallsimplerightR-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 otherradicals of a ring.The Jacobson radical is named afterNathan Jacobson,who was the first to study it for arbitrary rings inJacobson 1945.

Nathan Jacobson

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion tonon-unital rings.Theradical of a moduleextends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such asNakayama's lemma.

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 iscommutativeor not.

Commutative case

In the commutative case, the Jacobson radical of a commutative ringRis defined astheintersectionof allmaximal ideals${\mathfrak {m}}$.If we denoteSpecmRas the set of all maximal ideals inRthen

$\mathrm {J} (R)=\bigcap _{{\mathfrak {m}}\,\in \,\operatorname {Specm} R}{\mathfrak {m}}$

This definition can be used for explicit calculations in a number of simple cases, such as forlocal rings(R,${\mathfrak {p}}$),which have a unique maximal ideal,Artinian rings,andproductsthereof. See the examples section for explicit computations.

Noncommutative/general case

For a general ring with unityR,the Jacobson radical J(R) is defined as the ideal of all elementsr∈Rsuch thatrM= 0wheneverMis asimpleR-module. That is, $$\mathrm {J} (R)=\{r\in R\mid rM=0{\text{ for all }}M{\text{ simple}}\}.$$ This is equivalent to the definition in the commutative case for a commutative ringRbecause the simple modules over a commutative ring are of the formR/${\mathfrak {m}}$for some maximal ideal${\mathfrak {m}}$ofR,and theannihilatorsofR/${\mathfrak {m}}$inRare precisely the elements of${\mathfrak {m}}$,i.e.Ann_R(R/${\mathfrak {m}}$) =${\mathfrak {m}}$.

Motivation

Understanding the Jacobson radical lies in a few different cases: namely its applications and the resultinggeometricinterpretations, 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 inNakayama's lemma.This lemma is a technical tool for studyingfinitely generated modulesover commutative rings that has an easy geometric interpretation: If we have avector bundleE→Xover atopological spaceX,and pick a pointp∈X,then any basis ofE|_pcan be extended to a basis of sections ofE|_U→Ufor someneighborhoodp∈U⊆X.

Another application is in the case of finitely generated commutative rings of the form${\textstyle R=k[x_{1},\ldots,x_{n}]\,/\,I}$for some base ringk(such as afield,or the ring ofintegers). In this case thenilradicaland the Jacobson radical coincide. This means we could interpret the Jacobson radical as a measure for how far the idealIdefining the ringRis from defining the ring of functions on analgebraic varietybecause of theHilbert Nullstellensatztheorem. This is because algebraic varieties cannot have a ring of functions with infinitesimals: this is a structure that is only considered inscheme theory.

Equivalent characterizations

The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in manynoncommutative algebratexts such asAnderson & Fuller 1992,§15,Isaacs 1994,§13B, andLam 2001,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 allmaximal right idealsof the ring. The equivalence coming from the fact that for all maximal right idealsM,R/Mis a simple rightR-module, and that in fact all simple rightR-modules areisomorphicto one of this type via the map fromRtoSgiven byr↦xrfor any generatorxofS.It is also true that J(R) equals the intersection of all maximal left ideals within the ring.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 islocal,and has a unique maximalright 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.This symmetry stands in contrast to the lack of symmetry in thesoclesofR,for it may happen that soc(R_R) is not equal to soc(_RR). IfRis anon-commutative ring,J(R) is not necessarily equal to the intersection of all maximaltwo-sidedideals ofR.For instance, ifVis acountabledirect sum of copies of a fieldkandR= End(V)(thering of endomorphismsofVas ak-module), thenJ(R) = 0becauseRis known to bevon Neumann regular,but there is exactly one maximal double-sided ideal inRconsisting of endomorphisms with finite-dimensionalimage.
• J(R) equals the sum of allsuperfluous right ideals(or symmetrically, the sum of all superfluous left ideals) ofR.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 ofR;soc(R_R) is both the sum ofminimal right idealsand the intersection ofessential 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 allannihilatorsofsimplerightR-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 aprimitive 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, ifUis a rightR-module, andVis amaximal submoduleofU,U· J(R)is contained inV,whereU· J(R)denotes all products of elements of J(R) (the "scalars" ) with elements inU,on the right. This follows from the fact that thequotient moduleU/Vis simple and hence annihilated by J(R).
• J(R) is the unique right ideal ofRmaximal with the property that every element isright quasiregular(or equivalently left quasiregular). 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 lemmais perhaps the most well-known instance of this. Although every element of the J(R) is necessarilyquasiregular,not every quasiregular element is necessarily a member of J(R).
• While not every quasiregular element is in J(R), it can be shown thatyis in J(R)if and only ifxyis left quasiregular for allxinR.
• J(R) is the set of elementsxinRsuch that every element of1 +RxRis a unit:J(R) = {x∈R| 1 +RxR⊂R^×}.In fact,y∈Ris in the Jacobson radical if and only if1 +xyis invertible for anyx∈R,if and only if1 +yxis invertible for anyx∈R.This meansxyandyxbehave similarly to anilpotentelementzwithzⁿ⁺¹= 0and(1 +z)⁻¹= 1 −z+z²−... ±zⁿ.

Forrings without unityit is possible to haveR= J(R);however, the equationJ(R/ J(R)) = {0}still holds. The following are equivalent characterizations of J(R) for rings without unity:

• The notion of left quasiregularity can be generalized in the following way. Call an elementainRleftgeneralized quasiregularif there existscinRsuch thatc+a−ca= 0.Then J(R) consists of every elementafor whichrais left generalized quasiregular for allrinR.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 moduleMis amended by adding the condition thatR⋅M≠ 0.With this understanding, J(R) may be defined as the intersection of all annihilators of simple leftRmodules, or justRif there are no simple leftRmodules. Rings without unity with no simple modules do exist, in which caseR= J(R),and the ring is called aradical 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 integersZits Jacobson radical is thezero ideal,soJ(Z) = (0),because it is given by the intersection of every ideal generated by aprime number(p). Since(p₁) ∩ (p₂) = (p₁⋅p₂),and we are taking an infinite intersection with no common elements besides 0 between all maximal ideals, we have the computation.
• For alocal ring(R,${\mathfrak {p}}$)the Jacobson radical is simplyJ(R) =${\mathfrak {p}}$.This is an important case because of its use in applying Nakayama's lemma. In particular, it implies if we have an algebraic vector bundleE→Xover a scheme or algebraic varietyX,and we fix a basis ofE|_pfor some pointp∈X,then this basis lifts to a set of generators for all sectionsEfor some neighborhoodUofp.
• Ifkis afieldandR=k[[X₁,...,X_n]]is a ring offormal power series,then J(R) consists of thosepower serieswhose constant term is zero, i.e. the power series in the ideal(X₁,...,X_n).
• In the case of anArtinian rings,such asC[t₁,t₂]/(t₁⁴,t₁²t₂²,t₂⁹),the Jacobson radical is(t₁,t₂).
• The previous example could be extended to the ringR=C[t₂,t₃,...]/(t₂²,t₃³,...),givingJ(R) = (t₂,t₃,...).
• The Jacobson radical of the ringZ/12Zis 6Z/12Z,which is the intersection of the maximal ideals 2Z/12Zand 3Z/12Z.
• Consider the ringC[t] ⊗_CC[x₁,x₂]_{x1²+x2²−1},where the second is thelocalizationofC[x₁,x₂]by the prime ideal${\mathfrak {p}}$= (x₁²+x₂²− 1).Then, the Jacobson radical is trivial because the maximal ideals are generated by an element of the form(t−z) ⊗ (x₁²+x₂²− 1)forz∈C.

Noncommutative examples

• Rings for which J(R) is {0} are calledsemiprimitive rings,or sometimes "Jacobson semisimple rings". The Jacobson radical of any field, anyvon Neumann regular ringand any left or rightprimitive ringis {0}. The Jacobson radical of the integers is {0}.
• IfKis a field andRis the ring of allupper triangularn-by-nmatriceswith entries inK,then J(R) consists of all upper triangular matrices with zeros on the main diagonal.
• Start with a finite, acyclicquiverΓ and a fieldKand consider the quiver algebraK Γ (as described in the articleQuiver). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
• The Jacobson radical of aC*-algebrais {0}. This follows from theGelfand–Naimark theoremand the fact that for a C*-algebra, a topologically irreducible *-representation on aHilbert spaceis algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (seeSpectrum of a C*-algebra).

Properties

• IfRis unital and is not the trivial ring {0}, the Jacobson radical is always distinct fromRsincerings with unity always have maximal right ideals.However, some importanttheoremsandconjecturesin ring theory consider the case whenJ(R) =R– "IfRis a nil ring (that is, each of its elements isnilpotent), is thepolynomial ringR[x] equal to its Jacobson radical? "is equivalent to the openKöthe conjecture.
• For any idealIcontained in J(R),J(R/I) = J(R) /I.
• In particular, the Jacobson radical of the ringR/ J(R)is zero. Rings with zero Jacobson radical are calledsemiprimitive rings.
• A ring issemisimpleif and only if it isArtinianand its Jacobson radical is zero.
• Iff:R→Sis asurjectivering homomorphism,thenf(J(R)) ⊆ J(S).
• IfRis a ring with unity andMis afinitely generatedleftR-module withJ(R)M=M,thenM= 0(Nakayama's lemma).
• J(R) contains allcentralnilpotent elements, but contains noidempotent elementsexcept for 0.
• J(R) contains everynil idealofR.IfRis left or rightArtinian,then J(R) is anilpotent ideal.
  This can actually be made stronger: If
  {0} =T₀⊆T₁⊆ ⋅⋅⋅ ⊆T_k=R
  is acomposition seriesfor the rightR-moduleR(such a series is sure to exist ifRis right Artinian, and there is a similar left composition series ifRis left Artinian), then(J(R))^k= 0.
  Note, however, that in general the Jacobson radical need not consist of only thenilpotentelements of the ring.
• IfRis commutative and finitely generated as analgebraover either afieldorZ,then J(R) is equal to thenilradicalofR.
• 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

References

• Anderson, Frank W.; Fuller, Kent R. (1992),Rings and Categories of Modules,Graduate Texts in Mathematics,vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376,doi:10.1007/978-1-4612-4418-9,ISBN0-387-97845-3,MR1245487
• Atiyah, M. F.;Macdonald, I. G.(1969),Introduction to Commutative Algebra,Addison-Wesley Publishing Co., pp. ix+128,MR0242802
• Bourbaki, N.Éléments de mathématique.
• Herstein, I. N.(1994) [1968],Noncommutative Rings,Carus Mathematical Monographs, vol. 15, Washington, DC:Mathematical Association of America,pp. xii+202,ISBN0-88385-015-X,MR1449137Reprint of the 1968 original; With an afterword by Lance W. Small
• Isaacs, I. M.(1994),Algebra: a graduate course(1st ed.),Brooks/ColePublishing Company,ISBN0-534-19002-2
• Jacobson, Nathan(1945), "The radical and semi-simplicity for arbitrary rings",American Journal of Mathematics,67(2): 300–320,doi:10.2307/2371731,ISSN0002-9327,JSTOR2371731,MR0012271
• Lam, T. Y.(2001),A First Course in Noncommutative Rings,Graduate Texts in Mathematics, vol. 131 (2 ed.), Springer-Verlag, pp. xx+385,doi:10.1007/978-1-4419-8616-0,ISBN0-387-95183-0,MR1838439
• Pierce, Richard S. (1982),Associative Algebras,Graduate Texts in Mathematics, vol. 88, Springer-Verlag, pp.xii+436,ISBN0-387-90693-2,MR0674652Studies in the History of Modern Science, 9
• Smoktunowicz, Agata(2006), "Some results in noncommutative ring theory",International Congress of Mathematicians, Vol. II(PDF),European Mathematical Society,pp. 259–269,ISBN978-3-03719-022-7,MR2275597,archived fromthe original(PDF)on 2017-08-09,retrieved2014-12-31

External links

• Intuitive Example of a Jacobson Radical