First-year graduateAlgebraPlanned
Commutative algebra
Commutative rings and their modules: prime and maximal ideals, localisation, Noetherian rings and Hilbert's basis theorem, integral extensions, the Nullstellensatz, primary decomposition and dimension.
math.algebra.commutative-algebra
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
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
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)