Let $A$ be the ring of integers $\mathbb{Z}$ or the polynomial ring $\mathbb{C}[X]$. In each case, give an example of an ideal $I$ of $A$ such that the quotient ring $R=A / I$ has a non-trivial idempotent (an element $x \in R$ with $x \neq 0,1$ and $x^{2}=x$ ) and a non-trivial nilpotent element (an element $x \in R$ with $x \neq 0$ and $x^{n}=0$ for some positive integer $n$ ). Exhibit these elements and justify your answer.