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18.706 HOMEWORK 2

By Due Monday and March In Class

Abstract

Please type or print clearly. Write the exercise number on your paper (but you don’t have to rewrite the question). Justify your assertions. You may use [AM] as a reference. Changes I made to the original exercise are in bold face. Exercises from [FC]: 3.4 Show that the center of a simple ring is a field, and the center of a semisimple ring is a finite direct product of fields. 3.5 Let M be a left R-module and E = End(RM). If RM is a semisimple R-module, show that ME is a semisimple E-module. 3.7 Let R be a simple ring which is finite-dimensional over its center k. (By the way, this is the definition of a central simple algebra. k is a field by Exercise 4 above.) Let M be a finitely generated left R-module and let E = End(RM). Show that (dimk M) 2 = (dimk R)(dimk E). 3.14 (Over certain rings, the “rank ” of a free module may not be defined.) Le

Year: 2011
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