Ring and Ideal Proof

Prove properties of rings using ideals, quotient constructions, and the correspondence between ideals and quotients.

The Key Insight

Ideals are kernels of ring homomorphisms. The structure of R/I reveals properties of both R and I. Prime and maximal ideals correspond to domains and fields.

Step 1: Verify Ideal Properties

An ideal I of ring R must satisfy:

• (I, +) is a subgroup of (R, +)
• For all r ∈ R and a ∈ I: ra ∈ I and ar ∈ I (absorption)

Checking Ideal

To prove I is an ideal:

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1. 0 ∈ I: [show zero is in I]
2. Closure under +: If a, b ∈ I, then a + b = ... ∈ I
3. Additive inverses: If a ∈ I, then -a = ... ∈ I
4. Absorption: If r ∈ R and a ∈ I, then ra = ... ∈ I

Common Ideal Types

Notation	Definition	Generated by
(a)	Principal ideal	Single element a
(a, b)	Two-generated	Elements a and b
(a₁, ..., aₙ)	Finitely generated	Elements a₁, ..., aₙ
I + J	Sum	{a + b : a ∈ I, b ∈ J}
IJ	Product	{Σ aᵢbᵢ : aᵢ ∈ I, bᵢ ∈ J}
I ∩ J	Intersection	Set intersection

Step 2: Classify the Ideal

Prime Ideal

I is prime if I ≠ R and:

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ab ∈ I ⟹ a ∈ I or b ∈ I

Equivalent: R/I is an integral domain.

Maximal Ideal

I is maximal if I ≠ R and:

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I ⊆ J ⊆ R ⟹ J = I or J = R

Equivalent: R/I is a field.

Key Relationships

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Maximal ⟹ Prime (in commutative rings with 1)

I maximal ⟺ R/I is a field
I prime ⟺ R/I is an integral domain

Step 3: Work with Quotient Rings

Elements of R/I

The quotient ring R/I consists of cosets:

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R/I = { r + I : r ∈ R }

Operations in R/I

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(a + I) + (b + I) = (a + b) + I
(a + I) · (b + I) = (ab) + I

Zero Divisors

a + I is a zero divisor in R/I ⟺ ∃b ∉ I such that ab ∈ I.

Units

a + I is a unit in R/I ⟺ ∃b such that ab - 1 ∈ I ⟺ 1 ∈ (a) + I.

Step 4: Apply Ring Isomorphism Theorems

First Isomorphism Theorem

If φ: R → S is a ring homomorphism:

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R / ker(φ) ≅ im(φ)

Correspondence Theorem

Ideals of R/I correspond bijectively to ideals of R containing I:

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J/I ↔ J where I ⊆ J ⊆ R

Moreover:

•J/I is prime in R/I ⟺ J is prime in R
•J/I is maximal in R/I ⟺ J is maximal in R

Chinese Remainder Theorem

If I + J = R (coprime ideals):

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R / (I ∩ J) ≅ R/I × R/J

Step 5: Lean Formalization

lean

import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Quotient.Basic

variable {R : Type*} [CommRing R]

-- Ideal generated by single element
example (a : R) : Ideal R := Ideal.span {a}

-- Principal ideal definition
example (a : R) : Ideal.span ({a} : Set R) =
    { r | ∃ s, r = s * a } := by
  ext r
  simp [Ideal.mem_span_singleton]

-- Prime ideal definition
example (I : Ideal R) [I.IsPrime] (a b : R) (h : a * b ∈ I) :
    a ∈ I ∨ b ∈ I := Ideal.IsPrime.mem_or_mem ‹I.IsPrime› h

-- Maximal implies prime
example (I : Ideal R) [I.IsMaximal] : I.IsPrime :=
  Ideal.IsMaximal.isPrime ‹I.IsMaximal›

-- Quotient by maximal ideal is a field
example (I : Ideal R) [I.IsMaximal] : Field (R ⧸ I) :=
  Ideal.Quotient.field I

-- First isomorphism theorem for rings
-- RingHom.quotientKerEquivRange gives: R ⧸ ker(φ) ≃+* range(φ)

Example: Prove (p) is Maximal in ℤ

Problem: Show that for prime p, the ideal (p) = pℤ is maximal in ℤ.

Proof:

We show ℤ/(p) is a field.

The elements of ℤ/(p) are: {0̄, 1̄, 2̄, ..., p̄-1̄}

ℤ/(p) is an integral domain: If āb̄ = 0̄, then p | ab. Since p is prime and p | ab, we have p | a or p | b. Thus ā = 0̄ or b̄ = 0̄.

ℤ/(p) is a field: Since ℤ/(p) is a finite integral domain, it is a field. (Every nonzero element has finite multiplicative order, hence is a unit.)

Alternatively: For any ā ≠ 0̄, gcd(a, p) = 1 since p is prime. By Bezout: ∃s, t: sa + tp = 1, so s̄ā = 1̄, giving ā⁻¹ = s̄.

Since ℤ/(p) is a field, (p) is maximal. ∎

Example: Ideal Containment

Problem: In ℤ[x], show that (x, 2) ⊋ (x² + 2).

Proof:

We need to show (x² + 2) ⊆ (x, 2) and (x² + 2) ≠ (x, 2).

Containment: x² + 2 = x · x + 2 · 1 ∈ (x, 2) Therefore (x² + 2) ⊆ (x, 2).

Proper: Suppose (x, 2) = (x² + 2). Then x ∈ (x² + 2). So x = f(x) · (x² + 2) for some f(x) ∈ ℤ[x]. Comparing degrees: deg(x) = 1, but deg(f(x) · (x² + 2)) ≥ 2 for f ≠ 0. If f = 0, then x = 0, contradiction. Therefore (x, 2) ≠ (x² + 2), so (x² + 2) ⊊ (x, 2). ∎

Output Format

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**Claim:** [Statement about ideals/rings]

**Ideal verification:** (if proving something is an ideal)
[Check subgroup + absorption properties]

**Classification:** (if determining type)
[Check prime/maximal conditions]

**Quotient analysis:** (if using quotient)
R/I ≅ [identified ring]
[Properties of the quotient]

**Conclusion:**
[Final result]

**Lean Proof:**
[Formal verification]

Common Pitfalls

•Zero ring: The zero ring has (0) = R, which is both prime and maximal by some definitions but not others
•Non-commutative rings: Left ideals ≠ right ideals ≠ two-sided ideals
•Proper ideal: Prime and maximal require I ≠ R
•Containment direction: (a) ⊆ (b) ⟺ b | a in a PID
•Product vs. intersection: IJ ⊆ I ∩ J always, but equality requires conditions

Advanced Techniques

Localization

R_P (localization at prime P) makes elements outside P invertible:

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R_P = { r/s : r ∈ R, s ∉ P }

Unique maximal ideal: PR_P.

Nilradical and Jacobson Radical

•Nilradical: √(0) = ∩{P : P prime} = {a : aⁿ = 0 for some n}
•Jacobson radical: J(R) = ∩{M : M maximal}

Primary Decomposition

In Noetherian rings, every ideal is an intersection of primary ideals.