MT1 and Solutions.pdf

MT1 and Solutions.pdf - MATH 228(Q1 Mid-Term Examination 1...

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MATH 228 (Q1) - Mid-Term Examination 1 2 February 2018 12:00-12:50 Please place your ID card on the table for verification. No notes, textbooks, calculators, or other electronic or communication de- vices are allowed on the examination. There are FOUR questions on this examination. Q1. [5 points] Prove by induction on n that n + ( n + 1) + . . . + (2 n ) = 3 n ( n + 1) 2 for all n 1. Q2. [5 points] Let X be the subset of R consisting of all real numbers of the form a + b 7, where a, b are integers (this is the ring Z [ 7]). (a) Show that 0 X and 1 X (these are the usual integers zero and one). (b) Show that if x = a + b 7 X and y = m + n 7 X , then x - y X . (c) Show that if x = a + b 7 X and y = m + n 7 X , then xy X .
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Q3. [5 points] Let R be a commutative ring. (a) Define what it means for R to be an integral domain . (b) Define what it means for an element u R to be a unit of R . (c) Suppose R is an integral domain, let u be a unit of R . Suppose some element x R is such that x | u . Prove that x is a unit of R .
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  • Spring '12
  • Rahmati,Saeed
  • Math, Integral domain, Ring theory, Commutative ring, Unique factorization domain, Principal ideal domain

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