In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
ring theory
In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
, the Jacobson radical of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
consisting of those elements in
that
annihilate all
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
right
-
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
. 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
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
, who was the first to study it for arbitrary rings in .
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 In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle so ...
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 In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...
.
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
or not.
Commutative case
In the commutative case, the Jacobson radical of a commutative ring
is defined as
the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of all
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
s
. If we denote
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 ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
s
, which have a unique maximal ideal,
Artin rings, and
products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
thereof. See the examples section for explicit computations.
Noncommutative/general case
For a general ring with unity
, the Jacobson radical
is defined as the ideal of all elements
such that
whenever
is a
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
-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.
.
Motivation
Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
interpretations, and its algebraic interpretations.
Geometric applications
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
Nakayama's lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...
. It is a technical tool for studying
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts inclu ...
s over commutative rings which has an easy geometric interpretation: If we have a vector bundle
over a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, or the ring of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
). 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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
because of the
Hilbert Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
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
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
.
Equivalent characterizations
The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many
noncommutative algebra
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
texts such as , , and .
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to one of this type via the map from ''R'' to ''S'' given by ''r'' ↦ ''xr'' for any generator ''x'' of ''S''. It is also true that
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 is
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
, 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. This symmetry stands in contrast to the lack of symmetry in the
socles of ''R'', for it may happen that soc(''R''
''R'') is not equal to soc(
''R''''R''). If ''R'' is a
non-commutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...
,
is not necessarily equal to the intersection of all maximal ''two-sided'' ideals of ''R''. For instance, if ''V'' is a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
direct sum of copies of a field ''k'' and ''R'' = End(''V'') (the
ring of endomorphisms
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
of ''V'' as a ''k''-module), then
because
is known to be
von Neumann regular
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...
, but there is exactly one maximal double-sided ideal in ''R'' consisting of endomorphisms with finite-dimensional
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
.
*
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(''R''
''R'') 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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
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
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals ar ...
, 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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
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
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by t ...
''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 (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 lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...
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'').
* While not every quasiregular element is in
, it can be shown that ''y'' is in
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
''xy'' is left quasiregular for all ''x'' in ''R''.
*
is the set of elements ''x'' in ''R'' 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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
element ''z'' with ''z''
''n''+1 = 0 and
.
For
rings without unity it is possible for ''R'' = J(''R''); however, the equation J(''R''/J(''R'')) = 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 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.
Examples
Commutative examples
* For the ring of integers
its Jacobson radical is the
zero ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
, so
, because it is given by the intersection of every ideal generated by a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. Since
, and we are taking an infinite intersection with no common elements besides
between all maximal ideals, we have the computation.
* For a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and
is a ring of
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
, 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 In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
, 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
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of