Posts

Solutions manual for NTB - 1.2 Ideals and greatest common divisors

Image
Exercise 1.8. Let be a non-empty set of integers that is closed under addition (i.e., for all ). Show that is an ideal if and only if for all .

Proof

We prove is an ideal implies for all and vice versa.

First, we show is an ideal implies for all . Since is an ideal so we have for all . Choose , we get what we want to prove.

Now, we show for all implies is an ideal. For all , we have ( times ) which belongs to since is closed under addition.

Similarly, for all , we have ( times ) which belongs to since is closed under addtion and . So we have for all and for all .

This proves is an ideal.

(q.e.d.)



Exercise 1.9. Show that for all integers , we have:
(a)