Mwaka wa kwanza wa uzamiliAlgebraImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 29 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- Thinking every ring is Noetherian; k[x₁, x₂, x₃, …] is not.
- Confusing prime ideals with maximal ideals.
Inaelekea kwa
Ugunduzi wa hiari
- Homological algebra Regular local rings are characterised homologically, as the Noetherian local rings of finite global dimension (Auslander–Buchsbaum–Serre).
- ← Rings, ideals and fields
Njia za kujifunza
Usomaji zaidi
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (1969)
- David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (1995)