[PDF] definition of injective modules

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable  Examples · Commutative examples · Artinian examples · Modules over Lie algebrasAutres questions
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  • What are the example of injective modules?

    First examples
    Trivially, the zero module {0} is injective.
    Given a field k, every k-vector space Q is an injective k-module.
    Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V.
    The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K.

  • What is the product of injective modules?

    A direct product of injective modules is always injective.
    The corresponding property for direct sums does not hold in general, but it is true for modules over Noetherian rings.
    The notion of injective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors.

  • What is the injective resolution of a module?

    Injective resolutions are right resolutions whose Ci are all injective modules.
    Every R-module possesses a free left resolution.
    A fortiori, every module also admits projective and flat resolutions.

  • What is the injective resolution of a module?

    The minimal injective of R is a direct summand of Q and is therefore projective.
    We should note that if R is both left and right Noetherian and has a minimal faithful left module then the minimal injective of any pro- jective module is projective.

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