Consider the ring $\mathcal { C } ( \mathbb { R } )$ of continuous real-valued functions on $\mathbb { R }$, with pointwise addition and multiplication. For $A \subset \mathbb { R }$, the ideal of $A$ is $I ( A ) = \{ f \in \mathcal { C } ( \mathbb { R } ) \mid f ( a ) = 0$ for all $a \in A \}$. For a subset $I$ of $\mathcal { C } ( \mathbb { R } )$, the zero-set of $I$ is $Z ( I ) = \{ a \in \mathbb { R } \mid f ( a ) = 0$ for all $f \in I \}$. Prove the following: (A) (3 marks) $Z ( I \cap J ) = Z ( I J )$ for ideals $I$ and $J$ of $\mathcal { C } ( \mathbb { R } )$. (B) (2 marks) For each $a \in \mathbb { R } , I ( a )$ is a maximal ideal. (C) (3 marks) The set $\{ f \in \mathcal { C } ( \mathbb { R } ) \mid f$ has compact support $\}$ is a proper ideal, and its zero set is empty. (D) (2 marks) True/False: For each prime ideal $\mathfrak { p }$ of $\mathcal { C } ( \mathbb { R } ) , Z ( \mathfrak { p } )$ is a singleton set. (Justify your answer.)
Consider the ring $\mathcal { C } ( \mathbb { R } )$ of continuous real-valued functions on $\mathbb { R }$, with pointwise addition and multiplication. For $A \subset \mathbb { R }$, the ideal of $A$ is $I ( A ) = \{ f \in \mathcal { C } ( \mathbb { R } ) \mid f ( a ) = 0$ for all $a \in A \}$. For a subset $I$ of $\mathcal { C } ( \mathbb { R } )$, the zero-set of $I$ is $Z ( I ) = \{ a \in \mathbb { R } \mid f ( a ) = 0$ for all $f \in I \}$. Prove the following:\\
(A) (3 marks) $Z ( I \cap J ) = Z ( I J )$ for ideals $I$ and $J$ of $\mathcal { C } ( \mathbb { R } )$.\\
(B) (2 marks) For each $a \in \mathbb { R } , I ( a )$ is a maximal ideal.\\
(C) (3 marks) The set $\{ f \in \mathcal { C } ( \mathbb { R } ) \mid f$ has compact support $\}$ is a proper ideal, and its zero set is empty.\\
(D) (2 marks) True/False: For each prime ideal $\mathfrak { p }$ of $\mathcal { C } ( \mathbb { R } ) , Z ( \mathfrak { p } )$ is a singleton set. (Justify your answer.)