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  3. 9.2.6 exercise. let r = z and let i be the ideal 1

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i)...

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Advance Math

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R.

  1. (i) List explicitly all the ideals A of R with I ? A.
  2. (ii) Write out all the elements of R/I (these are cosets).
  3. (iii) List explicitly the set of all ideals B of R/I (these aresets of cosets).
  4. (iv) Let ?: R ? R/I be the natural projection. For each ideal Aof R such that I ? A, write out ?(A) explicitly (this is a set ofcosets). Confirm by direct calculation what the Third IsomorphismTheorem says: that the function A ?? ?(A) is a bijection from theset of such A that you found in step (i) and the set of ideals ofR/I that you found in step (iii).
  5. (v) For each ideal A of R with I ? A, write out all theelements of the following three quotient groups (under addition):R/A, A/I, and (R/I)/(A/I) (the last consists of cosets of cosets!).Then confirm by direct calculation what the Third IsomorphismTheorem says: that the rule f : R/A ? (R/I)/(A/I) with f(gA) =(gI)(A/I) makes a well defined function that is an isomorphism ofrings.

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