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In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective.
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Similarly, every injective R-module is an injective R-module.
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Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.
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For symmetric algebras, the duality is particularly well-behaved and projective modules and injective modules coincide.
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Every indecomposable injective module has a local endomorphism ring.
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Electronic fuel injection modules
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Fuel systems consisting of fuel injection modules and fuel injector for internal combustion engines
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Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3).
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Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in (Matlis 1958).
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Over a general integral domain, we still have one implication: every injective module over an integral domain is divisible.
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3.22); every infinite direct sum of injective modules is injective if and only if the ring is Noetherian, (Lam 1999, Th 3.46).
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The injective hull of a module is the smallest injective module containing the given one and was described in (Eckmann & Shopf 1953).
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For any Artinian ring, just as for commutative rings, there is a 1-1 correspondence between prime ideals and indecomposable injective modules.
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This is the Bass-Papp theorem, see (Papp 1959) and (Chase 1960) A module isomorphic to an injective module is of course injective.
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Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, (Lam 1999, §3F).
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Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields.
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An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases.
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Every module M also has an injective resolution: an exact sequence of the form 0 → M → I0 → I1 → I2 → ... where the I j are injective modules.
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Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category.
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For any injective right S-module M, the set of module homomorphisms HomS( P, M ) is an injective right R-module.
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The zero ideal is also prime and corresponds to the injective K. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.
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The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its injective hull.
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Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules.
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In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
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The factor group Q/Z and the circle group are also injective Z-modules.
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