Projective Module - Properties

Properties

• Direct sums and direct summands of projective modules are projective.
• If e = e2 is an idempotent in the ring R, then Re is a projective left module over R.
• Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called left hereditary.
• The category of finitely generated projective modules over a ring is an exact category. (See also algebraic K-theory).
• Every module over a field or skew field is projective (even free). A ring over which every module is projective is called semisimple.
• An abelian group (i.e. a module over Z) is projective if and only if it is a free abelian group. The same is true for all principal ideal domains; the reason is that for these rings, any submodule of a free module is free.
• Over a Dedekind domain a non-principal ideal is always a projective module that is not a free module.
• Over a direct product of rings R × S where R and S are nonzero rings, both R × 0 and 0 × S are non-free projective modules.
• Over a matrix ring M_n(R), the natural module Rn is projective but not free. More generally, over any semisimple ring, every module is projective, but the zero ideal and the ring itself are the only free ideals.
• Every projective module is flat. The converse is in general not true: the abelian group Q is a Z-module which is flat, but not projective.
• In line with the above intuition of "locally free = projective" is the following theorem due to Kaplansky: over a local ring, R, every projective module is free. This is easy to prove for finitely generated projective modules, but the general case is difficult.
• A finitely related module is flat if and only if it is projective.

The relation of projective modules to free and flat modules is subsumed in the following diagram of module properties:

Note that the last implication holds only for modules over a domain.