In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle...

years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently...

properties, a ring is semisimple if and only if it is Artinian and its Jacobson radical is zero. If an Artinian semisimple ring contains a field as a central...

In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more...

as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated...

that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal...

of all prime ideals of the quotient ring. This is contained in the Jacobson radical, which is the intersection of all maximal ideals, which are the kernels...

theory Radical of an ideal, an important concept in abstract algebra Radical of a ring, an ideal of "bad" elements of a ring Jacobson radical, consisting...

Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman (1951), who named them Hilbert...

algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional...

and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R). It is possible for a ring to have a unique maximal two-sided...

MR 0071721. Jacobson–Bourbaki theorem Jacobson's conjecture Jacobson density theorem Jacobson radical Jacobson ring "Nathan Jacobson (1910-1999)" (PDF)...

quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. In this article, we primarily concern ourselves with the...

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....

Nathan Jacobson (1910–1999), American mathematician Jacobson's conjecture Jacobson density theorem Jacobson radical Jacobson ring Norm Jacobson (1917–1994)...

abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian...

coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units...

prime ideal, the Jacobson radical — which is the intersection of maximal ideals — must contain the nilradical. A ring R is called a Jacobson ring if the nilradical...

observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the artinian hypothesis)...

but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0. A ring in which all elements are idempotent is called...

unnecessary. A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more...

necessarily vice versa. Jacobson 1.  The Jacobson radical of a ring is the intersection of all maximal left ideals. 2.  A Jacobson ring is a ring in which...

is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7) The above definition...

and Encryption, MIT Press, 1998, p. 108. Dissertation title: On the Jacobson Radical of a Group Algebra, see Ahmed Chalabi at the Mathematics Genealogy...

observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis)...

with the nilradical when commutativity is assumed. The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators...

by Φ ( G ) = G {\displaystyle \Phi (G)=G} . It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup...

ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical J = Jac ⁡ ( R ) {\displaystyle J=\operatorname {Jac} (R)} of R is the...

intersection of all prime ideals. A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent...

is Zariski with respect to 𝔞 if and only if 𝔞 is contained in the Jacobson radical of R. In particular a Noetherian local ring is Zariski with respect...