An injective module is the dual notion to the projective module. A module over a unit ring is called injective iff whenever is contained as a submodule in a module , there exists a submodule of such that the direct sum is isomorphic to (in other words, is a direct summand of ). The subset of is an example of a noninjective -module; it is a -submodule of , and it is isomorphic to ; , however, is not isomorphic to the direct sum . The field of rationals and its quotient module are examples 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.

See also

Baer's Criterion, Cofree Module, Commutative Diagram, Divisible Module, Exact Functor, Injective Object, Projective Module

This entry contributed by Margherita Barile

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References

Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, pp. 93-95, 1999.Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, pp. 87-91, 1998.Cartan, H. and Eilenberg, S. "Injective Modules." §1.3 in Homological Algebra. Princeton, NJ: Princeton University Press, pp. 8-10, 1956.Hilton, P. J. and Stammbach, U. "Dualization, Injective Modules" and "Injective Modules over a Principal Ideal Domain." §6 and 7 in A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, pp. 28-33, 1997.Jacobson, N. "Injective Modules. Injective Hull." §3.11 in Basic Algebra II. San Francisco, CA: W. H. Freeman, pp. 155-164, 1980.Lam, T. Y. "Injective Modules." §3 in Lectures on Modules and Rings. New York: Springer-Verlag, pp. 60-120, 1999.Lang, S. "Injective Modules." §20.4 in Algebra, rev. 3rd ed. New York: Springer-Verlag, pp. 782-786, 2002.Mac Lane, S. "Injective Modules." §7 in Homology. Berlin, Germany: Springer-Verlag, pp. 92-95, 1967.Northcott, D. G. "Injective Modules." §5.2 in An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, pp. 67-70, 1966.Passman, D. S. A Course in Ring Theory. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 206-210, 1991.Rowen, L. H. "Injective Modules." §2.10 in Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 261-270, 1988.Sharpe, D. W. and Vámos, P. Injective Modules. Cambridge, England: Cambridge University Press, 1972.

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Injective Module

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Weisstein, Eric W., with contributions by Margherita Barile. "Injective Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InjectiveModule.html

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