EDIT: I put the solution here in front so people don't have to wade through posts upon posts of confusion:
I just took a look at your book, and it seemed we can just substitute elements of whenever we want. In this case, our job is quite easy actually. To see that
is an ideal, we want to show that it is closed under addition by elements in and closed under multiplication by elements of . For the first bit, suppose . Then for all . Thus . For the second assertion, suppose . Then for all . Therefore .