In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 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 (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
''N'' of ''M'' is called
maximal or cosimple if the
quotient ''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, their ...
of all maximal submodules of ''M'',
:
Equivalently,
:
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.
In fact, if ''M'' is
finitely generated over a ring, then rad(''M'') itself is a superfluous submodule. This is because any proper submodule of ''M'' is contained in a maximal submodule of ''M'' when ''M'' is finitely generated.
* 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the
cosocle ''M''/rad(''M'') is finitely generated and rad(''M'') is a superfluous submodule of ''M''.
See also
*
Socle (mathematics)
*
Jacobson radical
References
*
*
Module theory
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