Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $R$ ?

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
I'm interested to know where I can find an explanation of why it is Dedekind. Thanks! – rschwieb Nov 13 '12 at 17:15
It's an interesting example that I had not seen in detail before. I imagine all commutative algebraists must be familiar with it! – rschwieb Nov 13 '12 at 22:03
@rschwieb, you can see it here,… 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 :-) – Ram Nov 14 '12 at 2:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.