modules over decomposable rings
Let be two, nontrivial, unital rings and . If is a -module and is a -module, then obviously is a -module via . We will show that every -module can be obtain in this way.
i.e. ring action on (respectively ) does not depend on (respectively ).
Proof. Let and . Of course both are idempotents and . Moreover and are central, i.e. . We will use to construct submodules . More precisely, let and . Because are central, then it is clear that both and are submodules. We will show that . Indeed, let . Then we have
Thus . Furthermore, assume that . Then there exist such that
Now, after multiplying both sides by we obtain that
thus . This shows that . To finish the proof, we need to show that the ring action on does not depend on (the other case is analogous). But this is clear, since for any and we have
This completes the proof.
|Title||modules over decomposable rings|
|Date of creation||2013-03-22 18:50:05|
|Last modified on||2013-03-22 18:50:05|
|Last modified by||joking (16130)|