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

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I learn the notion of "prime module" from wiki's article associated prime. According to wiki, it only needs to give a projective module which has no associated prime ideal.

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.

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    Ah, now I understand. Thanks for the explanation, @Michael.2012-08-12