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Jacobson radical

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Nathan Jacobson

In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-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 radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Template:Harvnb.

The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to non-unital rings. 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 R is defined as<ref>{{#invoke:citation/CS1|citation |CitationClass=web

}}</ref> the intersection of all maximal ideals <math>\mathfrak{m}</math>. If we denote Template:Nowrap as the set of all maximal ideals in R then

<math>\mathrm{J}(R) = \bigcap_{

\mathfrak{m} \,\in\, \operatorname{Specm}R

} \mathfrak{m}</math>

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

Noncommutative/general caseEdit

For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements Template:Nowrap such that Template:Nowrap whenever M is a 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 Template:Nowrap for some maximal ideal Template:Nowrap, and the annihilators of Template:Nowrap in R are precisely the elements of <math>\mathfrak{m}</math>, i.e. Template:Nowrap.

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

Template:See also 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 modules over commutative rings that has an easy geometric interpretation: If we have a vector bundle Template:Nowrap over a topological space X, and pick a point Template:Nowrap, then any basis of E|p can be extended to a basis of sections of Template:Nowrap for some neighborhood Template:Nowrap.

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, 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 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 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 Template:Harvnb, Template:Harvnb, and Template:Harvnb.

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 right ideals of the ring. The equivalence coming from the fact that for all maximal right ideals M, Template:Nowrap 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 Template:Nowrap for any generator x of S. It is also true that J(R) equals the intersection of all maximal left ideals within the ring.Template:Sfn 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 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.Template:SfnTemplate:Sfn 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, J(R) 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 Template:Nowrap (the ring of endomorphisms of V as a k-module), then Template:Nowrap 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.Template:Sfn
  • J(R) 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, J(R) 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, Template:Nowrap is contained in V, where Template:Nowrap 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 Template:Nowrap 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 quasiregularTemplate:SfnTemplate:Sfn (or equivalently left quasiregularTemplate:Sfn). 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).Template:Sfn
  • 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.Template:Sfn
  • J(R) is the set of elements x in R such that every element of Template:Nowrap is a unit: Template:Nowrap. In fact, Template:Nowrap is in the Jacobson radical if and only if Template:Nowrap is invertible for any Template:Nowrap, if and only if Template:Nowrap is invertible for any Template:Nowrap. This means xy and yx behave similarly to a nilpotent element z with Template:Nowrap and Template:Nowrap.

For rings without unity it is possible to have Template:Nowrap; however, the equation Template:Nowrap still holds. The following are equivalent characterizations of J(R) for rings without unity:Template:Sfn

  • 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 Template:Nowrap. 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 Template:Nowrap. 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 Template:Nowrap, 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

Noncommutative examplesEdit

PropertiesEdit

See alsoEdit

NotesEdit

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CitationsEdit

Template:Reflist

ReferencesEdit

External linksEdit

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