A projective module over a nonzero unit ring is a unital R-module that is a direct summand of a free module, i.e., of some direct sum . This does not necessarily imply that itself is a direct sum of copies of . A counterexample is provided by , which is a module over the ring with respect to the multiplication defined by . Hence, while a free module is obviously always projective, the converse does not hold in general. It is true, however, for particular classes of rings, e.g., if is a principal ideal domain. Every finitely generated module that is projective over a polynomial ring over a field is free (Quillen 1976, Suslin 1976). Since is a principal ideal domain, is a nonprojective -module, as it is not free.

A direct sum of projective modules is always projective, but this property does not apply to direct products. For example, the infinite direct product is not a projective -module.

Equivalently, a unital R-module is projective if every surjective module homomorphism from a unital R-module has a module homomorphism with for all . The specified module homomorphism then gives a direct sum , where is its kernel.

The notion of projective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors. It is dual to the notion of injective module.

See also

Algebraic K-Theory, Commutative Diagram, Faithfully Flat Module, Finitistic Dimension, Flat Module, Free Module, Injective Module, Projective Dimension, Projective Resolution, Serre's Problem

Portions of this entry contributed by Margherita Barile

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References

Cartan, H. and Eilenberg, S. "Projective Modules." §1.2 in Homological Algebra. Princeton, NJ: Princeton University Press, pp. 6-8, 1956.Hilton, P. J. and Stammbach, U. "Free and Projective Modules" and "Projective Modules over a Principal Ideal Domain." §4 and 5 in A Course in Homological Algebra, 2nd ed. New York: Springer-Verlag, pp. 22-28, 1997.Jacobson, N. "Projective Modules." §3.10 in Basic Algebra II. San Francisco, CA: W. H. Freeman and Company, pp. 148-155, 1980.Kunz, E. "Projective Modules." §3 in Introduction to Commutative Algebra and Algebraic Geometry. Boston, MA: Birkhäuser, pp. 110-112, 1985.Lam, T. Y. "Projective Modules." §2 in Lectures on Modules and Rings. New York: Springer-Verlag, pp. 21-59, 1999.Mac Lane, S. "Free and Projective Modules." in Homology. Berlin, Germany: Springer-Verlag, pp. 19-21, 1967.Northcott, D. G. "Projective Modules." §5.1 in An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, pp. 63-67, 1966.Passman, D. S. A Course in Ring Theory. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 18-20, 1991.Quillen, D. "Projective Modules over Polynomial Rings." Invent. Math. 36, 167-171, 1976. https://doi.org/10.1007/BF01390008.Rowen, L. H. "Projective Modules (An Introduction)." §2.8 in Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 225-237, 1988.Suslin, A. A. "Projective Modules over a Polynomial Ring Are Free." Dokl. Akad. Nauk SSSR 229, 1063-1066, 1976. https://www.mathnet.ru/eng/dan40545.

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Projective Module

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

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