Rings, ideals and fields

How many units does the ring $\Z/12\Z$ have?

Which of these rings are integral domains? (Select all that apply.)

Which of these subsets of $\Z[x]$ are ideals? (Select all that apply.)

$\Q[x]$ is a principal ideal domain. Which polynomial generates the ideal $I = (x^2 - 1,\; x^2 - 3x + 2)$?

In the proof that a Euclidean domain is a PID, we take a non-zero ideal $I$, choose $b \in I \setminus \{0\}$ of least size $\delta(b)$, and divide any $a \in I$ as $a = qb + r$ with $r = 0$ or $\delta(r) \lt \delta(b)$. Why must $r = 0$?

In $\Z[\sqrt{-5}]$ the element $3$ is irreducible. Which of these products show that $3$ is not prime — that is, $3$ divides the product but divides neither factor? (Select all that apply.)

How many Gaussian integers $x + yi$ have norm $x^2 + y^2 = 25$?

Which Lean statement formalises “every finite integral domain is a field”?

Write classify(n) returning a pair (units, zero_divisors) of sorted lists for the ring $\Z/n\Z$ ($n \ge 2$): units are the $a$ with $ab \equiv 1$ for some $b$, and zero_divisors the non-zero $a$ with $ab \equiv 0$ for some non-zero $b$. Then check that together they contain every non-zero residue exactly once.