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 ).
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