Mathematics.

rings

Ideals

Abstract Algebra II25 minDifficulty7 out of 10

You should know: rings

Overview

An ideal is a special subset of a ring that absorbs multiplication by any ring element, playing the same role for rings that normal subgroups play for groups: ideals are precisely the kernels of ring homomorphisms and the objects used to form quotient rings.

Formal Definition

Definition

Let R be a ring. A subset I ⊆ R is a (two-sided) ideal of R if it is an additive subgroup that absorbs multiplication:

(I,+)≤(R,+)(I, +) \leq (R, +)
I is an additive subgroup
∀r∈R, ∀x∈I: rx∈I  and  xr∈I\forall r \in R,\ \forall x \in I:\ r x \in I \ \text{ and } \ xr \in I
Absorption property
R/I={r+I:r∈R}R/I = \{ r + I : r \in R \}
Quotient ring formed from I

Properties

Principal ideal

(a)={ra:r∈R} is the ideal generated by a single element a.(a) = \{ ra : r \in R \} \text{ is the ideal generated by a single element } a.

Example: In ℤ, the ideal (3) = {..., -6, -3, 0, 3, 6, ...} = 3ℤ.

Kernel is an ideal

The kernel of any ring homomorphism φ:R→S is an ideal of R.\text{The kernel of any ring homomorphism } \varphi: R \to S \text{ is an ideal of } R.

Prime ideal

I is prime if ab∈I  ⟹  a∈I or b∈I (I≠R).I \text{ is prime if } ab \in I \implies a \in I \text{ or } b \in I \ (I \neq R).

Maximal ideal

I is maximal if I≠R and no ideal strictly lies between I and R; R/I is then a field.I \text{ is maximal if } I \neq R \text{ and no ideal strictly lies between } I \text{ and } R; \ R/I \text{ is then a field.}

Worked Examples

  1. 2ℤ is an additive subgroup of ℤ (sums and negatives of even integers are even).

    2Z≤(Z,+)2\mathbb{Z} \leq (\mathbb{Z}, +)

  2. Absorption: for any integer r and even x = 2k, rx = 2(rk) is even, so rx ∈ 2ℤ.

    r⋅2k=2(rk)∈2Zr \cdot 2k = 2(rk) \in 2\mathbb{Z}

  3. The quotient ring ℤ/2ℤ has two cosets, 0+2ℤ and 1+2ℤ, matching ℤ₂.

    Z/2Z≅Z2\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2

Answer: 2ℤ is an ideal of ℤ, and ℤ/2ℤ ≅ ℤ₂, which is even a field since 2 is prime and (2) is maximal.

Practice Problems

Difficulty 6/10

Is (4) a prime ideal of ℤ? Is it maximal? Justify.

Summary

  • An ideal I ⊆ R is an additive subgroup that absorbs multiplication by any ring element.
  • Ideals are exactly the kernels of ring homomorphisms, mirroring normal subgroups in group theory.
  • A prime ideal generalizes prime numbers; a maximal ideal produces a field as its quotient ring.
  • In ℤ, the ideal (n) is prime and maximal simultaneously precisely when n is a prime number.

References

  1. WebsiteWikipedia — Ideal (ring theory)