Commutative algebra

Which of these ideals are prime? (Choose all that apply.)

What is the Krull dimension of $\mathbb{Z}[x]$?

How many elements does the localisation $(\mathbb{Z}/360)_{(3)}$ have? (Here $(3)$ is the prime ideal of $\mathbb{Z}/360$ generated by $3$.)

In the proof of Hilbert's basis theorem, $L_d \subseteq R$ is the ideal of leading coefficients of degree-$d$ elements of $I \subseteq R[x]$. Why is there an $N$ with $L_N = L_{N+1} = L_{N+2} = \cdots$?

Let $R = \mathbb{Z}_{(p)}$, $\mathfrak{m} = p\mathbb{Z}_{(p)}$ and $M = \mathbb{Q}$. Then $\mathfrak{m}M = M$ but $M \neq 0$. Which hypothesis of Nakayama's lemma fails?

Let $I = (x^2 + y^2 - 1,\; x - 1) \subseteq \mathbb{C}[x, y]$: the unit circle and its tangent line at $(1, 0)$. Which of these polynomials lie in $I$ itself? (Choose all that apply.)

Which Lean statement formalises “the polynomial ring in infinitely many variables over a field $k$ is not Noetherian”?

The ideals of $\mathbb{Z}/n$ are $d\mathbb{Z}/n\mathbb{Z}$ for the divisors $d$ of $n$. Write prime_ideals(n) (for $n \ge 2$) returning the sorted list of divisors $d$ of $n$ for which the ideal generated by $d$ in $\mathbb{Z}/n$ is prime, checking the definition directly: proper, and $ab \in I \Rightarrow a \in I$ or $b \in I$ for all residues $a, b$.