MA554_JAN99 - Math 554 - Heinzer/Roche Qualifying Exam...

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Unformatted text preview: Math 554 - Heinzer/Roche Qualifying Exam January 1999 Name: (7) 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 /φ ( Z n ). Such a matrix A is called a presentation matrix of V . (15) 2. Consider the abelian group V = Z / (5 4 ) ⊕ Z / (5 3 ) ⊕ Z . (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 the element (5 2 , 5 , 5) in Z / (5 4 ) ⊕ Z / (5 3 ) ⊕ Z = V . Write down a presentation matrix for W . (3) Write down a presentation matrix for the quotient module V/W . (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 = ( w 1 ,...,w 5...
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MA554_JAN99 - Math 554 - Heinzer/Roche Qualifying Exam...

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