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...

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...

can be considered as a dual notion to that of the radical of a module. In set notation, s o c ( M ) = ∑ N  is a simple submodule of  M N . {\displaystyle...

in a ring R that annihilate all simple right R-modules Nilradical of a ring, a nilpotent ideal which is as large as possible Radical of a module, a component...

especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood...

denotes the intersection of all proper maximal submodules of M (the radical of the module), then the top of M is M/rad(M). In the case of local rings with maximal...

nonzero modules have maximal submodules, and also projective modules have maximal submodules. As with rings, one can define the radical of a module using...

of rings: Radical of a module Kaplansky radical Radical of a bilinear form Amitsur, S. A. (1952). "A general theory of radicals. I: Radicals in complete...

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. See also: Glossary of linear...

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...

Singular submodule Socle of a module Radical of a module Constructions Free module Quotient module Direct sum, Direct product of modules Direct limit, Inverse...

maximal semisimple quotient of the module. Socle Radical of a module Adolfo Ballester-Bolinches, Luis M. Ezquerro, Classes of Finite Groups, 2006, ISBN 1402047185...

definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex...

In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any...

flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a ring R...

is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R,...

the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense...

mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear...

operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a module or vector space...

mathematics, the Jacquet module is a module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation...

mathematical field of Lie theory, the radical of a Lie algebra g {\displaystyle {\mathfrak {g}}} is the largest solvable ideal of g . {\displaystyle {\mathfrak...

polymer chemistry, radical polymerization (RP) is a method of polymerization by which a polymer forms by the successive addition of a radical to building blocks...

In ring theory, a branch of mathematics, the radical of an ideal I {\displaystyle I} of a commutative ring is another ideal defined by the property that...

involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which...

radical rad(R) (Lam 1999, Example 3.41). A simple module is necessarily the socle of its injective hull. The injective hull of the residue field of a...

Jacobson radical. The theorem states that if R is a semiprimary ring and M is an R-module, the three module conditions Noetherian, Artinian and "has a composition...

calls the "populist radical right" starts with the idea of 'the nation'. He however rejects the use of nationalism as a core ideology of right-wing populism...

language of modules, see also Glossary of module theory. For specific types of algebras, see also: Glossary of field theory and Glossary of Lie groups...

class group Radical of an ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay...

Jacobson radical of R. R has a complete orthogonal set e1, ..., en of idempotents with each eiRei a local ring. Every simple left (right) R-module has a projective...