examples of ring, a maximum ideal and prime ideal

Question

1. Give some examples of ring and a maximal ideal such that the maximal ideal is not a prime ideal of the ring.
2. Give some examples of ring and a prime ideal such that the prime ideal is not a maximal ideal of the ring.

Maximal Ideal: Let R be a ring. A two-sided ideal I of R is called maximal if $I \neq R$ and no proper ideal of R properly contains I.

Prime Ideals Let R bea commutative ring. An ideal I of R is called prime if $I \neq R$ and whenever $ab\in I$ for elements a and b of R, either $a\in I$ or $b\in I$.

I know the definitions but cannot form the examples atleast two for each case. Please give me examples in elaborate form as far as possible.

Answer 1

Hint for 1: $R=2\mathbb{Z}$, $I=4\mathbb{Z}$.

Hint for 2: $R=\mathbb{Z}$, $I=\{0\}$.