(a) Let $I$ be an ideal of a commutative ring $R$ and assume $I \subseteq \bigcup_{i=1}^{n} P_{i}$ where the $P_{i}$ are prime ideals. Show that $I \subseteq P_{i}$ for some $i$.

(b) Show that $\left(x^{2}+1\right)$ is a maximal ideal of $\mathbb{R}[x]$. Show that the quotient ring $\mathbb{R}[x] /\left(x^{2}+1\right)$ is isomorphic to $\mathbb{C} .$

(c) For $a, b \in \mathbb{R}$, let $I_{a, b}$ be the ideal $(x-a, y-b)$ in $\mathbb{R}[x, y]$. Show that $I_{a, b}$ is a maximal ideal. Find a maximal ideal $J$ of $\mathbb{R}[x, y]$ such that $J \neq I_{a, b}$ for any $a, b \in \mathbb{R}$. Justify your answers.