R be the ring of real valued continuous functions on closed interval [0, 1] and I an ideal of R.

Question

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$.

Let $I$ be an ideal of $R$.

Assume $f(x) = \cos{(2πx)}$ and $g(x) = 2x$ both belong to $I$.

Does $h(x) = \sin{(πx)}$ belong to $I$?

Answer 1

Verify that $p:[0,1]\rightarrow\mathbb{R}$ that sends $x$ to $\frac{1}{(2x)^2+(cos(2\pi x))^2}$ is continuous. (Note that the denominator is never $0$ for any $x$) Also note that $p=\frac{1}{f^2+g^2}$. Since $I$ is an ideal and $f,g\in I$, we know that $f^2,g^2\in I$. Therefore $f^2+g^2\in I$. Since $p=\frac{1}{f^2+g^2}$ is an element of your ring. Thus, $(\frac{1}{f^2+g^2})(f^2+g^2)\in I$. Hence $1\in I$. Therefore the ideal $I$ is the whole ring...