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1. Commutative Algebra - Local Rings
Let R be any (possibly noncommutative) ring and let G be any group. Then the group ring R G is defined to be the set of formal linear combinations Sum;
http://crypto.stanford.edu/pbc/notes/commalg/local.xhtml

Extractions: We call a ring R local if R has exactly one maximal ideal M . In this case, we call A M the residue field of R . A ring with only finitely many maximal ideals is called semi-local Example: Any field F is local and F is its own residue field. Let R be any (possibly noncommutative) ring and let G be any group. Then the group ring R G is defined to be the set of formal linear combinations g G g g where every g R , and only finitely many g are nonzero, with componentwise addition: g g g g g g g and convolution product: g G g g h G h h k G g h k g h k Take the cyclic group of order 2 C x x . Then C x x . This is a local ring with maximal ideal x and its residue field is isomorphic to Let F be a field of characteristic p , that is sum i p , and let G be any abelian p -group , that is, the order of every element of G is a power of p . Then F G is local with unique maximal ideal M g g g with residue field isomorphic to F . 3. The ring a b a b b is local with residue field . The ring a b a b b b is semi-local. We can continue in this fashion: by taking the first n primes we can construct a semi-local ring with exactly n maximal ideals. 4. The ring

2. 2010 Research In Teams: Derived Category Methods In Commutative Algebra II | Ban
Derived category methods have proved to be very successful in ring theory, in particular in commutative algebra. Evidence is provided by cite{AJL97,bour10,
http://www.birs.ca/events/2010/research-in-teams/10rit158

Extractions: Lars Christensen (Texas Tech University) The book offers a systematic development of hyperhomological algebra. This includes the construction of the derived category of a general (associative) ring and a careful study of the functors of importance in ring theory. To demonstrate the strength and utility of the theory, and to motivate the choice of topics, the book includes an extensive course in central homological aspects of commutative ring theory. This part includes many recent results, which were discovered by means of derived category methods, and gives valuable new insight into the theory of commutative rings and their modules. Objective Authors Apart from the need for such a book, our motivation for actually writing it has two sources. It stems from repeated urgings from colleagues and from personal desires to see a presentation of the theory that not only makes the case for derived category methods but also provides a comprehensive and coherent reference for the theory and its applications in commutative algebra. The applicants maintain active research programs in commutative algebra. These programs rely heavily on derived category methods - and more advanced techniques of differential graded algebra

3. Introduction To Commutative Algebra | EHow.com
Sep 11, 2009 Introduction to Commutative Algebra. Algebra is simply logic. Algebraic reasoning is based on consistencies and rules that apply when