Algebra: Rings, Modules and Categories I by Carl Faith

By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and in addition, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok contains all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) contains similarity periods of easy crucial algebras, and for arbitrary commutative okay, this can be subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer team. ) various different cases of a marriage of ring idea and type (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, significantly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside of ring thought. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root for it's a corre­ spondence theorem for projective modules (Theorem four. 7) instructed by means of the Morita context. As a spinoff, this offers starting place for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the introduction.

Show description

Read Online or Download Algebra: Rings, Modules and Categories I PDF

Similar algebra & trigonometry books


The necessity for more advantageous arithmetic schooling on the highschool and faculty degrees hasn't ever been extra obvious than within the 1990's. As early because the 1960's, I. M. Gelfand and his colleagues within the USSR notion challenging approximately this related query and constructed a mode for offering easy arithmetic in a transparent and easy shape that engaged the interest and highbrow curiosity of millions of highschool and faculty scholars.

A Universal Construction for Groups Acting Freely on Real Trees

The idea of R-trees is a well-established and demanding quarter of geometric workforce concept and during this booklet the authors introduce a building that gives a brand new standpoint on workforce activities on R-trees. They build a gaggle RF(G), outfitted with an motion on an R-tree, whose components are convinced features from a compact actual period to the gang G.

Algebras, Representations and Applications: Conference in Honour of Ivan Shestakov's 60th Birthday, August 26- September 1, 2007, Maresias, Brazil

This quantity includes contributions from the convention on 'Algebras, Representations and functions' (Maresias, Brazil, August 26 - September 1, 2007), in honor of Ivan Shestakov's sixtieth birthday. This booklet can be of curiosity to graduate scholars and researchers operating within the thought of Lie and Jordan algebras and superalgebras and their representations, Hopf algebras, Poisson algebras, Quantum teams, team earrings and different themes

Extra resources for Algebra: Rings, Modules and Categories I

Example text

This order is called reverse, or dual, order. The ordered set (A, >*) is called the ordered set dual to A and is denoted by A *. Thus, the identity mapping 1A : A -+ A is a duality A -+ A *. The very simple fact of the existence of A * for each ordered set A has useful consequences which seem to belie the triviality of this fact. The most important of these is the duality principle which asserts that for every theorem about ordered sets there is a dual theorem obtained simply by reversing order.

2 Let 71, as usual, denote the set of integers. If a, b E 71, write a b or b > a. In the former case a = a V b and b = a /I. b. 4 Any "geometric" lattice is a lattice. By geometric lattice we mean a configuration, oriented as in the diagram, consisting of two classes of parallel lines that intersect each other.

If A and B are ordered sets, then a mapping I: A -'J>- B is an order homomorphism provided that a>b§t(a»t(b). 'if a, b EA. Then t is an order injection (resp. surjection, bijection) in case t is an order homomorphism and an injective (resp. surjective, bijective) mapping. An order isomorphism I: A -'J>- B is a bijection such that both I and 1-1 are order homomorphisms. Note that every order injection I: A ->- B induces an order isomorphism I: A -'J>- im t, hence, any bijective order homomorphism is an order isomorphism.

Download PDF sample

Rated 4.62 of 5 – based on 40 votes