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Math Help - Projective Modules

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    Projective Modules

    Prove that I_6 is simultaneously an injective and a projective module over itself.
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    MHF Contributor Drexel28's Avatar
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    Re: Projective Modules

    Quote Originally Posted by jcir2826 View Post
    Prove that I_6 is simultaneously an injective and a projective module over itself.
    As a \mathbb{Z}_6-module, \mathbb{Z}_6 is free. Surely you know that all free modules are simultaneously projective. Thus, all we have to show is that \mathbb{Z}_6 is injective. To do this we can use Baer's criterion, which says that \mathbb{Z}_6 will be injective over \mathbb{Z}_6 if and only if we can extend every \mathbb{Z}_6-homomorphism \mathfrak{a}\to\mathbb{Z}_6 can be extended to \mathbb{Z}_6\to\mathbb{Z}_6 for every \mathfrak{a} a left ideal of \mathbb{Z}_6. So...
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