Ideals, quotient rings, prime & maximal ideals, CRT

Prime vs maximal

For a commutative ring R with 1:

• I prime ⇔ R/I is an integral domain.
• I maximal ⇔ R/I is a field.
• maximal ⇒ prime; the converse fails: $(X) \subset \mathbb{Z}[X]$ is prime $(\mathbb{Z}[X]/(X) \cong \mathbb{Z}, a$ domain) but not maximal.
• In a PID, non-zero prime ⇔ maximal.

Standard quotients

Quotient	Is
$\mathbb{Z}[X]/(X)$	$\mathbb{Z}$
$\mathbb{Z}[X]/(X, p)$	𝔽_p, so (X, p) is maximal
$\mathbb{R}[X]/(X^{2} + 1)$	$\mathbb{C}$
$\mathbb{Z}[X]/(X^{2} + 1)$	$\mathbb{Z}[i]$
𝔽_p[X]/(f), f irreducible of degree n	𝔽$_{p^{n}}$
$\mathbb{Z}/n\mathbb{Z}$	field ⇔ n prime

Chinese remainder theorem

$\gcd(m, n) = 1 \Rightarrow \mathbb{Z}/$mn $\cong \mathbb{Z}/m \times \mathbb{Z}/n$ as rings. Consequences used constantly:

• Number of solutions of $x^{2} = 1$ in $\mathbb{Z}/105: 105 = 3\cdot5\cdot7$, two roots each $\Rightarrow 2^{3} = 8$. $- \varphi$ is multiplicative; the number of units is $\prod\varphi(p^{k})$. $- \mathbb{Z}/n$ is a product of local rings; idempotents correspond to factorisations.

Nilpotents, units, idempotents

• x nilpotent ⇒ 1 + x is a unit.
• The nilradical is the intersection of all prime ideals. $- \mathbb{Z}/n$ has non-trivial nilpotents ⇔ n is not squarefree.
• Fermat/Euler: $a^{\varphi(n)}$ ≡ 1 (mod n) for gcd(a, n) = 1; Wilson: (p−1)! ≡ −1 (mod p).
• Power sums: $\sum_{k=1}^{p-1}$ kᵐ ≡ −1 (mod p) if (p − 1) | m, and ≡ 0 otherwise.

Key takeaways

• Identify the quotient ring; prime/maximal follows instantly.
• Maximal ⇒ prime always; converse only in PIDs (and for non-zero ideals).
• CRT is the workhorse for every counting question modulo a composite.