# Projective Modules over the Ring of Trigonometric Functions

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.