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 $R$ ?
Projective Modules over the Ring of Trigonometric Functions
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abstract-algebra
commutative-algebra
ring-theory
projective-module
1 Answers
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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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1I'm interested to know where I can find an explanation of why it is Dedekind. Thanks! – 2012-11-13
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0It's an interesting example that I had not seen in detail before. I imagine all commutative algebraists must be familiar with it! – 2012-11-13
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0@rschwieb, you can see it here, http://math.colorado.edu/~ravi1033/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