103b_07s_1e - Math 103B Midterm Exam(50 points Friday...

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Math 103B Midterm Exam (50 points) Friday 5/4/2007 Please put your name and ID number on your blue book. CLOSED BOOK, but BOTH SIDES of one page of notes are allowed. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (18 pts.) Which are TRUE and which are FALSE? Do NOT give reasons. (a) x 5 + 9 x 2 + 3 is irreducible over Z [ x ]. (b) For any ring R , h a i = aR . (c) In an integral domain, every maximal ideal is prime. (d) Every finite integral domain is a field. (e) Every principal ideal domain is a unique factorization domain. (f) If ϕ : R S is a ring homomorphism, then { x | ϕ ( x ) = 0 } is an ideal of S . 2. (8 pts.) Let m,n,k be integers greater than 2. Let 1 be the unity of
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