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 ...

, in the theory of

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

s, the radical of a module is a component in the theory of structure and classification. It is a generalization of the

Jacobson radical In mathematics, more specifically ring 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 "left" in place of "right" in the definition yie ...

for

rings 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 ...

. In many ways, it is the dual notion to that of the socle soc(''M'') of ''M''.

Definition

Let ''R'' be 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 ...

and ''M'' a left ''R''-

. A

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...

''N'' of ''M'' is called maximal or cosimple if the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

''M''/''N'' is a

simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cycl ...

. The radical of the module ''M'' is 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 submodules of ''M'', :

\mathrm(M) = \bigcap_ N

Equivalently, :

\mathrm(M) = \sum_ S

These definitions have direct dual analogues for soc(''M'').

Properties

* In addition to the fact rad(''M'') is the sum of superfluous submodules, in a

Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the proper ...

rad(''M'') itself is a

superfluous submodule In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M ...

. * A ring for which rad(''M'') = for every right ''R''-module ''M'' is called a right V-ring. * For any module ''M'', rad(''M''/rad(''M'')) is zero. * ''M'' is a finitely generated module

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 ...

the cosocle ''M''/rad(''M'') is finitely generated and rad(''M'') is a superfluous submodule of ''M''.

References

* * Module theory {{Abstract-algebra-stub

Definition

Properties

See also

References