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MA554_JAN05

# MA554_JAN05 - QUALIFYING EXAMINATION Math 554 January 2005...

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QUALIFYING EXAMINATION Math 554 January 2005 - Prof. Ulrich 1. (12 points) Without proof give an answer to these questions: (a) For R a commutative ring and M an R -module, is the R -module Hom R ( M, R ) torsion free? (b) How many isomorphism classes are there of Z 20 -modules having exactly 625 elements? (c) How many isometry classes are there of alternating bilinear forms on a 3-dimensional vector space? 2. (15 points) Let R be a commutative ring and P an R -module. Prove that the following are equivalent: (a) For every R -linear map f : P N and every R -epimorphism π : M N there exists an R -linear map g : P M with πg = f . (b) There exists an R -module Q such that the direct sum P Q is a free R -module. 3. (15 points) Let R be an integral domain and F a free R -module with ordered basis { x 1 , . . . , x n } . Let M = Ru 1 + . . . + Ru n N = Rv 1 + . . . + Rv n be submodules of F with u i = j a ij x j , v i = j b ij x j ( a ij , b ij R ), and consider the n by n matrices A = ( a ij ), B = ( b ij ). (a) Prove that M and N are free R -modules if the determinant det A 6 = 0. (b) Assume det A 6 = 0. Prove that M = N if and only if det A and det B are associates.

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4. (13 points) Determine whether the matrices
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MA554_JAN05 - QUALIFYING EXAMINATION Math 554 January 2005...

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