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...

simple right R-modules Nilradical of a ring, a nilpotent ideal which is as large as possible Radical of a module, a component in the theory of structure and...

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...

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...

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

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...

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...

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

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,...

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...

example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms...

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...

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

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...

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...

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

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...

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

associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated...

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...

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...

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...

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...

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

equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials. Barbeau, E. J. (2003-10-09). Polynomials. Springer...

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...