In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are "close to zero". In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, the term ideal has multiple meanings. ...

Definition

The Jacobson radical is denoted by J(R) and can be defined in the following equivalent ways:

• the intersection of all maximal left ideals.
• the intersection of all maximal right ideals.
• the intersection of all annihilators of simple left R-modules
• the intersection of all annihilators of simple right R-modules
• the intersection of all left primitive ideals.
• the intersection of all right primitive ideals.
• { x ∈ R : for every r ∈ R there exists u ∈ R with u (1-rx) = 1 }
• { x ∈ R : for every r ∈ R there exists u ∈ R with (1-xr) u = 1 }
• if R is commutative, the intersection of all maximal ideals in R.
• the largest ideal I such that for all x ∈ I, 1-x is invertible in R

Note that the last property does not mean that every element x of R such that 1-x is invertible must be an element of J(R). Also, if R is not commutative, then J(R) is not necessarily equal to the intersection of all two-sided maximal ideals in R. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... Annihilators are a concept that occurs in ring theory, a branch of mathematics. ... In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... Annihilators are a concept that occurs in ring theory, a branch of mathematics. ... A left primitive ideal is the annihilator of a simple left module. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

The Jacobson radical is named for Nathan Jacobson, who first studied the Jacobson radical. Nathan Jacobson (October 5, 1910-December 5, 1999) was an American mathematician, who was born in Warsaw, Poland and went to America with his Jewish family in 1918. ...

Examples

• The Jacobson radical of any field is {0}. The Jacobson radical of the integers is {0}.
• The Jacobson radical of the ring Z/8Z (see modular arithmetic) is 2Z/8Z.
• 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.
• If K is a field and R = K[[X₁,...,X_n]] is a ring of formal power series, then J(R) consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring consists precisely of the ring's non-units.
• Start with a finite quiver Γ and a field K and consider the quiver algebra KΓ (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 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).

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed. ... In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed. ... C*-algebras are an important area of research in functional analysis. ... In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... The spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i. ...

Properties

• Unless R is the trivial ring {0}, the Jacobson radical is always an ideal in R distinct from R.

• If R is commutative and finitely generated, then J(R) is equal to the nilradical of R.

• The Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitive rings.

• If f : R → S is a surjective ring homomorphism, then f(J(R)) ⊆ J(S).

• If M is a finitely generated left R-module with J(R)M = M, then M = 0 (Nakayama lemma).

• J(R) contains every nil ideal of R. If R is left or right artinian, then J(R) is a nilpotent ideal. Note however that in general the Jacobson radical need not contain every nilpotent element of the ring.

• R is a semisimple ring if and only if it is Artinian and its Jacobson radical is zero.

In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ... In abstract algebra, a semiprimitive ring is a ring with zero Jacobson radical. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, Nakayamas Lemma is an important technical lemma in commutative algebra and algebraic geometry. ... Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. ... In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ... Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

See also

• Radical of a module
• Radical of an ideal

In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ...

References

• M.F. Atiyah, I.G. Macdonald. Introduction to Commutative Algebra.
• N. Bourbaki. Éléments de Mathématique.
• R.S. Pierce. Associative Algebras. Graduate Texts in Mathematics vol 88.
• T.Y. Lam. A First Course in Non-commutative Rings. Graduate Texts in Mathematics vol 131.

This article incorporates material from Jacobson radical on PlanetMath, which is licensed under the GFDL. Introduction to Commutative Algebra is a commutative algebra textbook written by M. F. Atiyah and I. G. Macdonald. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...