# Algebra I: Rings, Modules, and Categories by Carl Faith

By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating rules of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy outcome, and additionally, a similarity category [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring okay comprises 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) involves similarity periods of easy primary algebras, and for arbitrary commutative ok, this can be subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer staff. ) a variety of different cases of a marriage of ring thought and class (albeit a shot­ gun wedding!) are inside the textual content. in addition, in. my try and extra simplify proofs, particularly to get rid of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside ring conception. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre­ spondence theorem for projective modules (Theorem four. 7) steered by way of the Morita context. As a spinoff, this offers starting place for a slightly whole thought of straightforward Noetherian rings-but extra approximately this within the creation.

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Extra resources for Algebra I: Rings, Modules, and Categories

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The clue at location M also leads to the prize. You are in charge of the scavenger hunt and want to sort out this information. (a) Draw a directed graph to represent the scavenger hunt. (b) How many paths are there to the prize (assuming we never visit a location twice)? (c) Identify a shortest path to the prize. 3. In the grid below, you can move any adjacent square into the blank square (effectively moving the blank square either horizontally or vertically). 11, each square can be given coordinates relative to the top left corner; for example, (3, 3) denotes the current position of the blank square, which we would like to move to (2, 1).

3: A more challenging maze not want to use trial-and-error. We will develop a more systematic and general approach. Our key strategy is to represent a maze as a graph, composed of vertices and edges. The next example illustrates these concepts in the context of social networks. 1 Represent a Social Network as a Graph A group of five students is selected from a large lecture class to work on a class project. A few of them are close friends with one another, while others are not. Specifically, Anand is friends with Brittany and with Claire; Dexter is also friends with Brittany and Claire; in addition, Claire is a friend of Ethan.

A) Using a graph representation, determine if it is possible to do this. (b) If so, describe the sequence of steps to measure out exactly 6 minutes. How long does it take? 27. We have egg timers that measure out 7 minutes and 4 minutes, respectively. A recipe requires that we be able to measure out exactly 5 minutes. (a) Using a graph representation, determine if it is possible to do this. (b) If so, describe the sequence of steps to measure out exactly 5 minutes. How long does it take? 28. A baker in ancient Greece has two hourglasses.