MA554_AUG03

MA554_AUG03 - QUALIFYING EXAM Math 554 August 2003 1. (12...

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QUALIFYING EXAM Math 554 August 2003 1. (12 points) Without proof give an example of: (a) a domain R and a finitely generated torsionfree R -module that is not free; (b) a nonzero commutative ring R that is not a field so that every R -module is torsionfree; (c) a normal matrix with entries in R that is not similar to a diagonal matrix with entries in R . 2. (13 points) For R a commutative ring and M an R -module let M ? =Hom R ( M,R ) denote the R -dual of M . (a) Let M and N be R -modules. Show that ( M N ) ? = M ? N ? . (b) Let R be a principal ideal domain and M a finitely generated R -module. Show that M ? is a free R -module with rank M ? =rank M . 3. (12 points) Let Q [ X ] be the polynomial ring in one variable over Q , n a positive integer, and R = Q [ X ] / ( X n ). Classify all finitely generated R -modules up to R -isomorphisms. 4. (15 points) Let R be a commutative ring, F afree R -module of finite rank, and ϕ End R ( F )an R -endomorphism of F . Show that the following are equivalent: (a) ϕ is bijective; (b) ϕ is surjective; (c) det( ϕ ) is a unit of R . 5. (20 points) Let K be a field, V a K -vector space of dimension n ,End K ( V ) the K -vector space of K -endomorphisms of V ,and ϕ End K ( V )afixed K - endomorphism. Show that:
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MA554_AUG03 - QUALIFYING EXAM Math 554 August 2003 1. (12...

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