Integral Domains

Prove that the ring

is an integral domain.

Let $n$ be a composite number. Prove that $\mathbb{Z}_{n}$ is not an integral domain.

Is the trivial ring an integral domain?

What is the characteristic of $\mathbb{Z}_{2}[\alpha]$ in 1.3?

Find two zero divisors in the ring

Let the integer $n$ be such that $\sqrt{|n|}$ is not an integer. Let the ring

1. Show that $R$ is an integral domain.

2. Is $R$ a field for all integers $n$?