स्नातकोत्तर प्रथम वर्षAlgebraनियोजित
Modules
Modules over rings, submodules and quotients, exact sequences, free, projective and injective modules, tensor products, and the structure theorem for finitely generated modules over a principal ideal domain.
math.algebra.modules
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
28 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Work with module homomorphisms, kernels, cokernels and exact sequences.
- Prove the structure theorem for finitely generated modules over a PID, and deduce the classification of finitely generated abelian groups and the Jordan normal form.
- Compute tensor products of modules and describe their effect on exact sequences.
परिभाषा
- A module over a ring R is an abelian group with an action of R satisfying the distributive and associative laws.
- A sequence A → B → C is exact at B if the image of the first map equals the kernel of the second.
- A module P is projective if Hom(P, −) carries surjections to surjections.
प्रमेय
- A finitely generated module over a PID R is isomorphic to a direct sum R/(d₁) ⊕ ⋯ ⊕ R/(dₖ) with d₁ | d₂ | ⋯ | dₖ.
- Free modules are projective, and over a PID every submodule of a free module is free.
- M ⊗_R − is right exact but in general not exact.
आम भ्रांतियाँ
- Expecting modules to behave like vector spaces: ℤ/2ℤ is a ℤ-module with no basis.
- Assuming tensoring preserves injectivity: multiplication by 2 on ℤ is injective, but becomes zero after tensoring with ℤ/2ℤ.
आगे ले जाता है
वैकल्पिक खोजें
- Adjunctions The tensor–hom adjunction Hom(M ⊗ N, P) ≅ Hom(M, Hom(N, P)), natural in all three variables, is a basic example of an adjunction.
अधिगम पथ
आगे पढ़ने के लिए
- David S. Dummit and Richard M. Foote, Abstract Algebra
- Serge Lang, Algebra