Can you help me to prove:
1:The ideal $(i, | i |)$is not principal.($i$ is identity function)
[ If $(i, | i |) = (d)$, there exists $g, h \in C(\mathbb{R})$(the real Continuous functions ) so that $i =gd$ and $ |i| = hd$ .It follows that $ g(0) = h(0) =0 $,morever there exist $s, t \in C(\mathbb{R})$ so that $ si + t |i| = d$. this implies that $ sg + th = 1$(constant function), a contradiction.]
2:Exhibite a principal ideal containing $(i, | i |)$.