Derived from
|
20402094 AL410 - COMMUTATIVE ALGEBRA in Mathematics LM-40 N0 TARTARONE FRANCESCA
(syllabus)
1. Modules Modules and submodules. Operations between submodules. Omomorphisms and quotient modules. Generators and bases. Free modules. Invariance of rank. Direct sum and direct product. Tensor product of modules. Universal property. Tensor product of algebras. Exactness of tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. The Nakayama Lemma.
2. Ideals Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson radical and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime Avoidance Theorem. Fractional ideals of domains. Invertible ideals.
3. Rings and fraction modules Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules. Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings. Local properties. Ring of formal series on a field.
4. Integral dependence Integral dependence and integral closure. Properties of stability and transitivity of integral dependence. Lying over, Inc and Going up. Krull dimension of the integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of Krull on integralclosure. Dedekind rings
5. Noetherian and Artinian rings and modules. Chain conditions and equivalent properties. Noetherian and Artininan rings. Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem. Primary decomposition of ideals. Uniqueness theorems. Prime associates and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings The Principal Ideal Theorem.
(reference books)
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972
|