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Let $ R = \mathbb{R}[ \cos x, \sin x] $ and consider the ideal $ \langle 1 - \cos x, \sin x\rangle $. Is this ideal a projective module over $R$ ?

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The ring $\mathbb{R}[\cos x, \sin x]$ is isomorphic to $\mathbb{R}[X,Y]/(X^2+Y^2-1)$ which is known as being a Dedekind domain, so all ideals are projective.

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    @rschwieb, you can see i$t$ here, h$t$$t$p://math.colorado.edu/~ravi1$0$33/notes/pre08/pdf/real_polynomials_on_unit_circle_dedekind.pdf I already read that before this post, expecting some straight forward solution, but on your reply I found that it is again the same complex manipulations :-)2012-11-14