MA554_AUG08 - MATH 554 August 2008 Instructions: Give a...

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Unformatted text preview: MATH 554 August 2008 Instructions: Give a complete solution to each question. For problems with multiple parts you may assume the result of the previous parts to solve the subsequent parts. Begin each problem on a new sheet of paper. Be sure your name is on every sheet of your solutions Notation: The following are standard for this examination. If R is a ring, M n ( R ) is the collection of n n matrices with R-entries, and R [ x ] is the ring of plynomials with R coefficients. The symbols Z , Q , R , and C denote the integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. The symbol I n denotes the n n identity matrix, and I V is the identity transformation of a vector space V. 1. (10 points) Let R be a principal ideal domain. A finitely generated R-module M is said to be indecomposable if no submodule of M is a direct summand of M, i.e., it is impossible to find proper submodules M 1 ,M 2 of M so that M = M 1 M 2 ....
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MA554_AUG08 - MATH 554 August 2008 Instructions: Give a...

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