prime ideal but not maximal ideal and maximal ideal but not prime ideal

Question

(i) With justification, give an example of a ring R and a prime ideal A of R such that A is not a maximal ideal in R.

(ii) With justification, give an example of a ring R and a maximal ideal B of R such that B is not a prime ideal in R.

I know the definitions of Prime and Maximal ideals

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 ideal: Let R be a 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 $b \in I$.

Answer 1

For a commutative ring with unity $R$ and an ideal $I$, we have the following:

1. $I$ is prime if and only if $R/I$ is an integral domain.
2. $I$ is maximal if and only if $R/I$ is a field.

This shows that every maximal ideal is a prime ideal since all fields are integral domains (every unit is not a zero-divisor).

The isomorphism $R[x]/(x) \cong R$ also provides a way to construct a prime ideal which is not maximal. For instance, pick any polynomial ring over a domain which is not a field and look at the ideal $(x)$. In particular, $\mathbb{Z}[x]$ for instance, $(x)$ is prime but not maximal since $\mathbb{Z} \cong \mathbb{Z}[x]/(x)$ is an integral domain, but not a field.

If the ring does not have an identity, it is actually possible. For instance $2 \mathbb{Z}$, the even integers. The ideal $(4)$ is maximal but not prime since $2\cdot 2 \in (4)$ but $2\notin (4)$.