finite dimensional modules over algebra
Assume that is a field, is a -algebra and is a -module over . In particular is a -module and a vector space over , thus we may speak about being finitely generated as -module and finite dimensional as a vector space. These two concepts are related as follows:
Proposition. Assume that and are both unital and additionaly is finite dimensional. Then is finite dimensional vector space if and only if is finitely generated -module.
Proof. ,,” Of course if is finite dimensional, then there exists basis
Thus every element of can be (uniquely) expressed in the form
which is equal to
,,” Assume that is finitely generated -module. In particular there is a subset
such that every element of is of the form
with all . Let be with the decomposition as above. Now is finite dimensional, so there is a subset
which is a -basis of . In particular for each we have
with . Thus we obtain
which shows, that all together make a set of generators of over (note that and are independent on ). Since it is finite, then is finite dimensional and the proof is complete.
|Title||finite dimensional modules over algebra|
|Date of creation||2013-03-22 19:16:35|
|Last modified on||2013-03-22 19:16:35|
|Last modified by||joking (16130)|