Homework Hint Needed:
Let $R$ be the ring of all continuous functions on $\mathbb{R}$ of period $\pi$. Let $S$ be the $R$-module of all continuous functions on $\mathbb{R}$ of antiperiod $\pi$. So elements of $S$ satisfy $$f(x + \pi) = -f(x).$$
How can I show that this module is not cyclic?
It seems that it has something to do with not being able to write both sine and cosine as multiples of a single antiperiodic function $p(x)$. How would I go about doing this?
Say $p$ is a continuous antiperiodic function that is cyclic. Note any multiple of $p$ must be zero wherever $p$ is zero. Since there exist anti periodic functions with disjoint zero sets (e.g. sine and cosine), what must be true about the zero set of $p$? Does there exist such a $p$?