Commutative algebra

Builds on

Required prerequisites, all the way down: 29

You will learn to

• Localise rings and modules, and use localisation to reduce problems to local rings.
• Prove Hilbert's basis theorem and Nakayama's lemma.
• State and use the Nullstellensatz to relate ideals and algebraic sets.

Definition

• The spectrum Spec R is the set of prime ideals of R.
• A ring is Noetherian if every ascending chain of ideals stabilises.
• The localisation S⁻¹R formally inverts the elements of a multiplicative subset S.

Theorem

• Hilbert's basis theorem: if R is Noetherian then so is R[x].
• Nullstellensatz: over an algebraically closed field k, the maximal ideals of k[x₁, …, xₙ] correspond to the points of kⁿ.
• Nakayama's lemma for finitely generated modules over local rings.
• The Krull dimension of k[x₁, …, xₙ] is n.

Common misconceptions

• Thinking every ring is Noetherian; k[x₁, x₂, x₃, …] is not.
• Confusing prime ideals with maximal ideals.

Leads to

Optional discoveries

• Homological algebra Regular local rings are characterised homologically, as the Noetherian local rings of finite global dimension (Auslander–Buchsbaum–Serre).
• ← Rings, ideals and fields

Learning pathways

• Graduate preparation
• Algebra and number theory

Further reading

• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (1969)
• David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (1995)