Jacobson radical of a module category and its power
Assume that is a field and is a -algebra. The category of (left) -modules will be denoted by and will denote the set of all -homomorphisms between -modules and . Of course is an -module itself and is a -algebra (even -algebra) with composition as a multiplication.
Let and be -modules. Define
Definition. The Jacobson radical of a category is defined as a class
Properties. The Jacobson radical is an ideal in , i.e. for any , for any , any and any we have and . Additionaly is an -submodule of .
For any -module we have , where on the right side we have the classical Jacobson radical.
In particular, if and are not isomorphic, then .
Let and let . Assume there is a sequence of -modules and for any we have an -homomorphism such that . Then we will say that is -factorizable through Jacobson radical.
Definition. The -th power of a Jacobson radical of a category is defined as a class
where is an -submodule of generated by all homomorphisms -factorizable through Jacobson radical. Additionaly define
Properties. Obviously and for any we have
Of course each is an -submodule of and we have following sequence of inclusions:
If both and are finite dimensional, then there exists such that
|Title||Jacobson radical of a module category and its power|
|Date of creation||2013-03-22 19:02:23|
|Last modified on||2013-03-22 19:02:23|
|Last modified by||joking (16130)|