HW1B - Math103B Homework Solutions HW1 Jacek Nowacki April...

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Unformatted text preview: Math103B Homework Solutions HW1 Jacek Nowacki April 26, 2007 Chapter 13 Problem 8. Describe all zero-divisors and units of R = Z Q Z . Proof. All elements of R are triplets of the form ( z 1 , q, z 2 ), where z 1 , z 2 Z and q Q . Now, we note that both operations (addition and multiplication) are component wise. If r R is a zero-divisor, it means that there is a q R such that rq = 0 and q 6 = 0(it is clear that R is commutative. if it was not, then we would have to be a little more careful). As elements of R , we can write r = ( z 1 , q, z 2 ) and q = ( z 1 , q , z 2 ). rq = 0 tells us that in fact: z 1 z 1 = 0, qq = 0, and z 2 z 2 = 0. We know that the only zero-divisor in both Q and Z is 0. So, if all components of r are not equal to 0, then all comonents of q must be zero. That is a contradiction. Hence, we get that the set of all zero-divisor is { r = ( z 1 , q, z 2 ) | r 6 = 0 and at least one component is 0 } ....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

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HW1B - Math103B Homework Solutions HW1 Jacek Nowacki April...

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