TITLE: Is every submodule of a projective module projective?
QUESTION [10 upvotes]: Is every submodule of a projective module projective?  

I know that the answer is no, but I haven't been able to come up with any concrete examples despite quite a bit of effort.
 Also, if the question is changed to whether or not every submodule of a free module is projective,what happens then? Any suggestions would be appreciated.

REPLY [3 votes]: Let $k$ be a field. The ideal
$$(x_1, x_2) =x_1 k[x_1,x_2] + x_2 k[x_1, x_2]\subset k[x_1, x_2]$$
is not a projective $k[x_1, x_2]$ module. 
It's enough to check that $(x_1, x_2)$ is not invertible see http://en.wikipedia.org/wiki/Fractional_ideal#Definition_and_basic_results .
Remark: In fact that every finitely generated projective module over $k[x_1, \ldots, x_n]$ must be free : http://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem