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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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