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

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

(i) List explicitly all the ideals A of R with I ⊆ A.
(ii) Write out all the elements of R/I (these are cosets).
(iii) List explicitly the set of all ideals B of R/I (these are sets of cosets).
(iv) Let π: R → R/I be the natural projection. For each ideal A of R such that I ⊆ A, write out π(A) explicitly (this is a set of cosets). Confirm by direct calculation what the Third Isomorphism Theorem says: that the function A ?→ π(A) is a bijection from the set of such A that you found in step (i) and the set of ideals of R/I that you found in step (iii).
(v) For each ideal A of R with I ⊆ A, write out all the elements 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 Isomorphism Theorem 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 of rings.

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Colby Messinger answered 2 years ago

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

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