product of injective modules is injective
Proof. Let be an arbitrary -module, a submodule and a homomorphism. It is enough to show that can be extended to . For denote by the projection. Since is injective for any , then the homomorphism can be extended to . Then we have
It is easy to check, that if , then , so is an extension of . Thus is injective.
Remark. Unfortunetly direct sum of injective modules need not be injective. Indeed, there is a theorem which states that direct sums of injective modules are injective if and only if ring is Noetherian. Note that the proof presented above cannot be used for direct sums, because need not be an element of the direct sum, more precisely, it is possible that for infinetly many . Nevertheless products are always injective.
|Title||product of injective modules is injective|
|Date of creation||2013-03-22 18:50:17|
|Last modified on||2013-03-22 18:50:17|
|Last modified by||joking (16130)|