113 HW 9_Math 113

# 113 HW 9_Math 113 - Mathematics 113 Homework 9 Curtis...

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Unformatted text preview: Mathematics 113: Homework # 9 Curtis Tekell Dr. Poirier November 8, 2010 Exercise 1: Determine all ideals of Z / 12 Z , which are prime, and which are maximal. Proof. Z is a PID, hence the ideals of Z are ( n ) for any positive n ≥ 0. The ideals in Z / 12 Z are all ideals of Z containing (12), or containing 12, that is, (1) = Z / 12 Z , (2), (3), (4), (6), and (12) = (0). Of these, (2) and (3) are the maximal ideals, and therfore prime. There are no other prime ideals. Exercise 2: (i) Describe the ring structure on Z × Z . (ii) Describe all ring homomorphisms f : Z × Z → Z . Proof. (i) The ring addition and multiplication are componentwise. The identity is 1 × 1. (ii) It is clear that Z × Z is free on X = { 1 × 1 , × 1 } , hence for any map f : X → Z , there is a unique homomorphism ˜ f : Z × Z → Z , extending f . As Z is an integral domain, if f is nonzero, f (1 × 1) = 1 for any homomorphism f , so then, the homomorphisms f are completely determined by f (0 × 1)....
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113 HW 9_Math 113 - Mathematics 113 Homework 9 Curtis...

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