modules over algebars and homomorphisms between them
Let be a ring and let be an associative algebra (not necessarily unital).
Definition. A (left) -module over is a pair where is a (left) -module and
is a -bilinear map such that the following conditions hold:
for any , and . We will simply use capital letters to denote modules.
Let be an -module over . If and then by we denote -submodule of generated by elements of the form for and . We will call unitary if . Note, that if has multiplicative identity , then is unitary if and only if for any .
The reason we use name ,,-module over ” instead of ,,-module” is that these to concepts may differ. The latter means that we treat simply as a ring and take modules over it. But such module need not be equiped with a ,,good” -module structure. On the other hand this is always the case, when is unitary over unital algebra.
It can be easily checked that -modules over together with -homomorphisms form a category which is abelian. Furthermore, if is unital, then its full subcategory consisting unitary -modules over is equivalent to category of unitary -modules.
In most cases it is important to assume that the base ring is a field, even algebraically closed.
|Title||modules over algebars and homomorphisms between them|
|Date of creation||2013-03-22 19:16:32|
|Last modified on||2013-03-22 19:16:32|
|Last modified by||joking (16130)|