स्नातकोत्तर प्रथम वर्षAlgebraपाठ उपलब्ध
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
इस पर आधारित
29 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- Thinking every ring is Noetherian; k[x₁, x₂, x₃, …] is not.
- Confusing prime ideals with maximal ideals.
आगे ले जाता है
वैकल्पिक खोजें
- Homological algebra Regular local rings are characterised homologically, as the Noetherian local rings of finite global dimension (Auslander–Buchsbaum–Serre).
- ← Rings, ideals and fields
अधिगम पथ
आगे पढ़ने के लिए
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra (1969)
- David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (1995)