Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$.
1) Prove that $I$ is an ideal.
2) What is $R \setminus I$ isomorphic to?
I have hints for two possible solutions but I am unable to reach a conclusion, there is always one assumption that I cannot verify.
Solution 1: prove by the definitions that I is an ideal. Use a relation between $f^2(0) = - f^2(1)$ to derive a relation between f(0) and f(1) (we can do this since we are working in $\mathbb{R}$. Use this to prove the "closed under addition" property of $I$. Then I am unable to prove that $\forall r \in R$ and $\forall f \in I$ that $f r$ and $r f$ are in $I$.
Any hints?
Assuming we now know that $I$ is an ideal, I now show that it is isomorphic to $\mathbb{R}^{+}$ by using the isomorphism theorem for rings: We exhibit the homomorphism $\phi : R \rightarrow \mathbb{R}^{+}$ such that $\forall f \in R, \phi(f) = f^2(0) + f^2(1)$. Now this is the part where I am also stuck, I don't know if this is a homomorphism, I cannot verify the simple definition.
Any hints?
If I can prove the latter, I is the kernel of $\phi$ so we know that $R\setminus I$ is isomorphic to $\mathbb{R}^{+}$.
Solution 2: this solution uses exclusively the isomorphism theorem for rings with the same homomorphism $\phi$ discussed here above. The final ring $\mathbb{R}^{+}$ is a ring so $\lbrace 0 \rbrace$ is an ideal. By the theorem we can construct a mapping from the ideals of $\mathbb{R}^{+}$ and the ideals of R with the following mapping: for $W'$ ideal in $\mathbb{R}^{+}$ we associate an ideal $W$ in R such that $W = \left\lbrace f \in R \mid \phi(f) \in W'\right\rbrace$. By taking $W' = \lbrace 0 \rbrace$ we just proved that $W$ is an ideal but such $W$ is nothing but $I$: $I$ is an ideal of R.
The second part of solution 2 is the same as for the first solution.
Thank you for your help.
Hints
$a^2+b^2=0$, with $a,b\in\mathbb{R}$ is equivalent to $a=b=\dots$
Consider the mapping $\varphi$ defined on $R$ with values in $\mathbb{R}\times\mathbb{R}$ given by $\varphi(f)=(f(0),f(1))$