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Projective Module


A projective module generalizes the concept of the free module. A module M over a nonzero unit ring R is projective iff it is a direct summand of a free module, i.e., of some direct sum  direct sum _IR. This does not imply necessarily that M itself is the direct sum of some copies of R. A counterexample is provided by M=Z, which is a module over the ring R=Z direct sum Z with respect to the multiplication defined by (a direct sum b)·x=ax. 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 R is a principal ideal domain, or a polynomial ring over a field (Quillen and Suslin 1976). This means that, for instance, Q is a nonprojective Z-module, since 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 Z×Z×... is not a projective Z-module.

According to its formal definition, a module M is projective if, whenever M is a quotient of a module N, there exists a module X such that the direct sum M direct sum X is isomorphic to N (in other terms, M is a direct summand of N).

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

Commutative diagram, Problem, Faithfully Flat Module, Flat Module, Free Module, Injective Module, Serre's Problem

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.Kunz, E. "Projective Modules." §3 in Introduction to Commutative Algebra and Algebraic Geometry. Boston, MA: Birkhäuser, pp. 110-112, 1985.Jacobson, N. "Projective Modules." §3.10 in Basic Algebra II. San Francisco, CA: W. H. Freeman and Company, pp. 148-155, 1980.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: 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.Rowen, L. H. "Projective Modules (An Introduction)." §2.8 in Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 225-237, 1988.

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Projective Module

Cite this as:

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

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