In mathematics, in the 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 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 definitio ...

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

Definition

Let ''R'' be a ring and ''M'' a left ''R''- module. 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 ...

''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 ...

. 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, thei ...

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 pr ...

rad(''M'') itself is a superfluous submodule. * 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 bi ...

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