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.
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: Springer-Verlag, pp. 92-95, 1967.Passman, D. S. A
Course in Ring Theory. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 206-210,
1991.Northcott, D. G. "Injective Modules." §5.2
in An
Introduction to Homological Algebra. Cambridge, England: Cambridge University
Press, pp. 67-70, 1966.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.