Jacobson radical

In mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -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 or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in (Jacobson 1945).

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. 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.

DefinitionsEdit

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 or not.

Commutative caseEdit

In the commutative case, the Jacobson radical of a commutative ring   is defined as[1] the intersection of all maximal ideals  . If we denoted   as the set of all maximal ideals in   then

 

This definition can be used for explicit calculations in a number of simple cases, such as for local rings  , which have a unique maximal ideal, Artin rings, and products thereof. See the examples section for explicit computations.

Noncommutative/general caseEdit

For a general ring with unit  , the Jacobson radical   is defined as the ideal of all elements   such that   whenever   is a simple module. That is,

 

This is equivalent to the definition in the commutative case for a commutative ring   because the simple modules over a commutative ring are of the form   for some maximal ideal  , and the only annihilators of   in   are in  , i.e.  .

MotivationEdit

Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations.

Geometric applicationsEdit

Although Jacobson originally introduced his radical as a technique for build 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 Nakayama's lemma. It is a technical tool for studying finitely generated modules over commutative rings which has an easy geometric interpretation: If we have a vector bundle   over a topological space  , and pick a point  , then any basis of   can be extended to a basis of sections of   for some neighborhood  .

Another application is in the case of finitely generated commutative rings, meaning   is of the form

 

for some base ring   (such as a field, or the ring of integers). In this case the nilradical and the Jacobson radical coincide. This means we could interpret the Jacobson radical as a measure for how far the ideal   defining the ring   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 which is only considered in scheme theory.

Equivalent characterizationsEdit

The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many noncommutative algebra texts such as (Anderson 1992, §15), (Isaacs 1994, §13B), and (Lam 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):

  •   equals the intersection of all maximal right ideals of the ring. The equivalence coming from the fact that for all maximal right ideals M, 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 rxr for any generator x of S. It is also true that   equals the intersection of all maximal left ideals within the ring.[2] 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, and has a unique maximal right ideal, then this unique maximal right ideal is exactly  . 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  . The left-right symmetry of these two definitions is remarkable and has various interesting consequences.[2][3] This symmetry stands in contrast to the lack of symmetry in the socles of R, for it may happen that soc(RR) is not equal to soc(RR). If R is a non-commutative ring,   is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V is a countable direct sum of copies of a field k and R = End(V) (the ring of endomorphisms of V as a k-module), then   because   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. (Lam 2001, p. 46, Ex. 3.15)
  •   equals the sum of all 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(RR) is both the sum of minimal right ideals and the intersection of essential right ideals. In fact, these two relationships hold for the radicals and socles of modules in general.
  • As defined in the introduction,   equals the intersection of all annihilators of 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, U·J(R) is contained in V, where 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 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 right quasiregular[4][5] (or equivalently left quasiregular[2]). 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, not every quasiregular element is necessarily a member of J(R).[5]
  • While not every quasiregular element is in  , it can be shown that y is in   if and only if xy is left quasiregular for all x in R. (Lam 2001, p. 50)
  •   is the set of elements xR such that every element of 1 + RxR is a unit:  . In fact,   is in the Jacobson radical if and only if 1 + xy is invertible for any  , if and only if 1+ yx is invertible for any  . This means xy and yx behave similarly to a nilpotent element z with zn+1 = 0 and  .

For rings without unity it is possible for R = J(R); however, the equation J(R/J(R)) = {0} still holds. The following are equivalent characterizations of J(R) for rings without unity (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 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 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 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.

ExamplesEdit

Commutative examplesEdit

  • For the ring of integers   its Jacobson radical is the zero ideal, so  , because it is given by the intersection of every ideal generated by a prime number  . Since  , and we are taking an infinite intersection which no common elements besides   between all maximal ideals, we have the computation.
  • For a local ring   the Jacobson radical is simply  . 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   over a scheme or algebraic variety  , and we fix a basis of   for some point  , then this basis lifts to a set of generators for all sections   for some neighborhood   of  .
  • If   is a field and   is a ring of formal power series, then   consists of those power series whose constant term is zero, i.e. the power series in the ideal  .
  • In the case of an Artin ring, such as  , the Jacobson radical is  .
  • The previous example could be extended to the ring  , giving  .
  • The Jacobson radical of the ring Z/12Z is 6Z/12Z, which is the intersection of the maximal ideals 2Z/12Z and 3Z/12Z.
  • Consider the ring   where the second is the localization of   by the prime ideal  . Then, the Jacobson radical is trivial because the maximal ideals are generated by an element of the form   for  .

Noncommutative examplesEdit

  • Rings for which J(R) is {0} are called semiprimitive rings, or sometimes "Jacobson semisimple rings". The Jacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. The Jacobson radical of the integers is {0}.
  • If K is a field and R is the ring of all upper triangular n-by-n 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 Γ and a field K and consider the quiver algebra K Γ (as described in the Quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
  • The Jacobson radical of a C*-algebra is {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).

PropertiesEdit

J(R / I) = J(R) / I.[6]
This can actually be made stronger: If
 
is a 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
 .
(Proof: Since the factors   are simple right R-modules, right multiplication by any element of J(R) annihilates these factors.
In other words,
 ,
whence
 .
Consequently, induction over i shows that all nonnegative integers i and u (for which the following makes sense) satisfy
 .
Applying this to u = i = k yields the result.)
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 either a field or Z, then J(R) is equal to the nilradical of R.
  • The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.

See alsoEdit

NotesEdit

  1. ^ "Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-24.
  2. ^ a b c Isaacs 1994, p. 182.
  3. ^ Isaacs 1994, Problem 12.5, p. 173.
  4. ^ Isaacs 1994, Corollary 13.4, p. 180.
  5. ^ a b Isaacs 1994, p. 181.
  6. ^ Lam (2001, §4, Prop. 4.6)

ReferencesEdit

External linksEdit