What is prime ideal?

Example 2.5

In $\Z$: the prime ideals are $(0)$ and the $(p)$, $p$ prime; the maximal ones are the $(p)$ ($\Z/p\Z = \mathbb F_p$ is a field, $\Z/(0) = \Z$ is not). In $K[X, Y]$: $(X) \subsetneq (X, Y)$ are both prime ($K[X,Y]/(X) \cong K[Y]$, a domain; $K[X,Y]/(X,Y) \cong K$, a field), so $(X)$ is prime but not maximal.

Example 2.10

In $\Z$ with $I_k = (m_k)$, $m_k$ pairwise coprime: $\Z/(m_1\cdots m_n)\Z \cong \prod \Z/m_k\Z$ — the Year 2 volume’s Chinese remainder theorem. Restricting to units: $(\Z/mn\Z)^\times \cong (\Z/m\Z)^\times \times (\Z/n\Z)^\times$ for $\gcd(m,n)=1$, whence the multiplicativity of Euler’s $\varphi$ (Exercise 2.8).