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ahw7 - Math 5125 Monday October 17 Seventh Homework...

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Math 5125 Monday, October 17 Seventh Homework Solutions 1. Note that n Z m Z because m n . By the ideal correspondence theorem we now obtain Z / n Z m Z / n Z = Z / m Z . However the ideals m Z / n Z and m ( Z / n Z ) of Z / n Z are equal, because they both con- sist of the cosets mr + n Z for some r Z , and the result follows. 2. Section 7.3, Exercise 35 on page 250. Let I , J , K be ideals of R . (a) Prove that I ( J + K ) = IJ + IK and ( I + J ) K = IK + JK . (b) Prove that if J I , then I ( J + K ) = J +( I K ) . (a) Obviously IJ , IK I ( J + K ) and since I ( J + K ) is closed under addition, we see that IJ + IK I ( J + K ) . On the other hand the general element of I ( J + K ) is of the form r i r ( j r + k r ) , where i r I , j r J and k r K , and this can be written in the form r i r j r + r i r k r , which shows that I ( J + K ) IJ + IK and it follows that I ( J + K ) = IJ + IK . The proof that ( I + J ) K = IK + JK is exactly similar. (b) Obviously J I (given), J J + K , I K I and I K J + K . Since I ( J + K ) is closed under addition, we deduce that J +( I K ) I ( J + K ) . Now suppose
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