Seminar on Commutative Algebra

[Motif]

Seminar on Commutative Algebra
这个Seminar奠基于Atiyah和Macdonald的经典文献《Intrduction to Commutative Algebra》，同时参考了Matsumura的《Commutative Algebra》（Second Edition）。AMB（Atiyah and Macdonald book）体例为：Chap 1:Rings and Ideals;Chap 2:Modules;Chap 3:Rings and Modules of Fractions;Chap 4:Primary Decomposition;Chap 5:Integral Dependence and Valuations;Chap 6:Chain conditions:Chap 7:Noetherian Rings;Chap 8:Artin Rings;Chap 9:Discrete Valuation Rings and Dedekin Domains;Chap 10Completions;Chap 11:Dimension Theory.
Let's begin.Chap 1,some basic definitions.What's a ring? A ring A is a set with two binary operations (addition and multiplication)such that:
I) A is an abelian group with respect to sddition;
II) Multiplication is associative and distributive over addition;
We shall consider only rings which are commutative:
III) xy=yx for all x,y ∈A;
and have an identity element(denoted by 1)
IV)∃1∈A such that x1=1x=x for all x∈A.
Remark:Throughout this seminar the word "ring" shall mean a commutative ring with an identity element ,that is , a ring satifying axioms (I) to (IV) above.
Rings Homomorphisms:
A ring homomorphism is a mapping f of a ring A into a ring B such that:
1) f(x+y)=f(x)+f(y);
2) f(xy)=f(x)f(y);
3) f(1)=1;
Ideals:
An ideal α of a ring A is a subset of A which is an additive subgroup and is such that A α ⊆ α(i.e.,x ∈A and y∈α imply xy∈α)。
Quotient Rings :
The quotient group A/α inherits a uniquely defined multiplication from A which makes it into a ring ,called the quotient ring(or residue-class ring ) A/α. The elements of A/α are the cosets of α in A ,and the mapping Φ:A→A/α which maps each x ∈A to its coset x+α is a surjective ring homomorphism。
做商是一种重要的技术，或许这个概念源自于拓扑学里的商空间，有时做商能够“模掉”一些不好的性质，从而使得所获的商结构有一些更加优美的性质，同时，又可“遗传”下母结构的部分性质。

[yijun]

要说明取ideal和取模的重要性，不妨提前给出一些由ideal或模来刻画的结构性定理。

[Motif]

1, I.S.Cohon's theorem: If all the prime ideals of a ring A are finitely generated then A is Noetherian.
proof:cf 《commutative ring theory 》by Matsumura,page 17.
2, In commutative ring theory,according to Matsumura,we have three top theorem in order of importance.The important three thoerems are: Krull's dimension theorem, I.S.Cohon's structure theorem for complete local rings, Serre's characterisation of a regular local ring.