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MA554_AUG95 - Math 554 Qualifying Exam August 1995 Name(10...

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Math 554 Qualifying Exam August 1995 Name: (10) 1. Let V be an abelian group and assume that ( v 1 , . . ., v m ) are generators of V . Describe a process for obtaining an m × n matrix A Z m × n such that if φ : Z n Z m is the Z -module homomorphism defined by left multiplication by A , then V = Z m ( Z n ). Such a matrix A is called a presentation matrix of V . (15) 2. Consider the abelian group V = Z / (5 3 ) Z / (5 2 ) Z / (5 2 ). (1) Write down a presentation matrix for V as a Z -module. (2) Let W be the cyclic subgroup of V generated by the image of (10 , 2 , 1) in Z / (5 3 ) Z / (5 2 ) Z / (5 2 ) = V . Write down a presentation matrix for W . (3) Write down a presentation matrix for the quotient Z -module V/W . 1
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(20) 3. Let R be a commutative ring and let V and W denote free R -modules of rank 4 and 5, respectively. Assume that φ : V W is an R -module homomorphism, and that B = ( v 1 , . . ., v 4 ) is an ordered basis of V and B 0 = ( w 1 , . . ., w 5 ) is an ordered basis of W . (1) What is meant by the coordinate vector of v V with respect to the basis B ?
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