A projective module generalizes the concept of the free module. A module over a nonzero unit ring is projective iff
it is a direct summand of a free
module, i.e., of some direct sum. This does not imply necessarily that itself is the direct sum of
some 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, or a polynomial
ring over a field (Quillen and Suslin 1976). This means that, for instance, is a nonprojective -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 is not a projective -module.
According to its formal definition, a module is projective if, whenever is a quotient of a module , there exists a module such that the direct sum is isomorphic to (in other terms, is a direct summand of ).
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.
over a nonzero unit ring 
is projective iff
it is a direct summand of a free
module, i.e., of some direct sum 
. This does not imply necessarily that 
itself is the direct sum of
some 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, or a polynomial
ring over a field (Quillen and Suslin 1976). This means that, for instance, 
is a nonprojective 
-module, since it is not free.
is not a projective 
-module.
is projective if, whenever 
is a quotient of a module 
, there exists a module 
such that the direct sum 
is isomorphic to 
(in other terms, 
is a direct summand of 
).
