Rings, ideals and fields

Builds on

Required prerequisites, all the way down: 15

You will learn to

• Verify the ring axioms and find ideals, units and zero divisors in examples such as ℤ, ℤ/nℤ, ℤ[i] and k[x].
• Prove that Euclidean domains are principal ideal domains and that principal ideal domains are unique factorisation domains.
• Show that ℤ[√−5] is not a unique factorisation domain.

Definition

• An ideal of a commutative ring R is an additive subgroup I such that ra ∈ I for all r ∈ R and a ∈ I.
• An integral domain is a commutative ring with 0 ≠ 1 and no zero divisors.
• A principal ideal domain is an integral domain in which every ideal is generated by a single element.

Theorem

• Euclidean domain ⇒ principal ideal domain ⇒ unique factorisation domain, and neither implication reverses.
• If k is a field, the polynomial ring k[x] is a Euclidean domain.
• Every finite integral domain is a field.
• The kernels of ring homomorphisms are exactly the two-sided ideals.

Common misconceptions

• Assuming every ring is commutative or has a multiplicative identity; conventions differ, so state them.
• Confusing subrings with ideals.
• Assuming irreducible and prime elements coincide in every integral domain.

Leads to

Optional discoveries

• Commutative algebra Ideals of polynomial rings correspond to geometric shapes cut out by equations, the starting point of commutative algebra and algebraic geometry.Wonder Trail →
• ← Whole-number arithmetic
• ← Factors, multiples and primes
• ← Integers
• ← Variables and algebraic expressions
• ← Divisibility, Euclid's algorithm and unique factorisation
• ← Congruences

Learning pathways

• Undergraduate mathematics major
• Graduate preparation

Further reading

• David S. Dummit and Richard M. Foote, Abstract Algebra
• Michael Artin, Algebra (1991)