# An example of a primeless (i.e. module without prime submodule) and projective module

Please, give an example of a module $M$ such that $M$ is primeless (i.e. without prime submodule) and projective. Thanks for your attention.

-

Let $A=k[x_1,x_2,\ldots]/(x_i^2\mid i=1,2,\ldots)$ which is a quotient ring of the infinitely many indeterminates polynomial ring over a field $k$. Then $A$ has only one prime ideal $\mathfrak{m}=(x_1,x_2,\ldots)$. It is not an associated prime, you cannot find an $f\in A$, such that $(0:f)=\mathfrak{m}$. And $A$ is a free module as $A$-module.
@m. sam. : You wrote: "Thanks for your answer, but I am looking for a module $M$ such that $M$ is not a free or a multiplication module and moreover $M$ is primeless and projective." The proper place for such a comment is down here in the comments section. I've deleted it from the answer. –  Michael Hardy May 27 '12 at 18:09