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mt2sols

mt2sols - Math 250A Fall 2004 Last Midterm ExamNovember 4...

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Math 250A, Fall 2004 Last Midterm Exam—November 4, 2004 Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Explain your answers in full English sentences as is customary and appropriate. Your paper is your ambassador when it is graded. All rings are rings with identity! 1. Suppose that I and J are ideals in a commutative ring R such that I + J = R . Establish the surjectivity of the natural map R R/I × R/J, r ( r mod I, r mod J ) . (Don’t just name a theorem; write down a complete proof.) This is the Chinese Remainder Theorem, but we are asked to supply a proof. If I + J = R , then 1 is an element of I + J , so that there are x I , y J with x + y = 1. Given a, b R , we can write down r := ay + bx and see that r has the same image as a mod I and the same image as b mod J . If I + J = R , as above, show that I n + J m = R whenever n and m are positive integers. We can take m = n since I n + J m contains I n + J n if n m . Suppose that 1 = x + y as above. Then 1 = ( x + y ) 2 n . If you expand out ( x + y ) 2 n by the binomial theorem, you’ll see that each term is divisible either by x n or by y n . Hence ( x + y ) 2 n lies in I n + J n . 2. Let k be a field, and let V be the k -vector space consisting of ( a 1 , a 2 , . . . ) with a i k and a i non- zero only for a finite set of i . Let R = End k V be the ring of linear transformations V V ; the ring multiplication is composition. (If V were instead the smaller vector space

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mt2sols - Math 250A Fall 2004 Last Midterm ExamNovember 4...

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