PUMaC 2023 · 加试 · 第 1 题

Problem 1.1.4. Show that the set of odd integers, as a subset of Z , is not an ideal. (Hint: which property does this set not satisfy?) Sometimes, we want to specify an ideal of R without having to explicitly list all of the elements. In particular, we only need to specify a subset of the elements of the ideal, knowing that our ideal satisfies the properties in the definition. For instance, note that any ideal that contains 2 also must contain the even integers. Indeed, note that if 2 lies in an ideal I, then so does every even integer, since each even integer is equal to 2 times some other integer. As the even integers are an ideal, it thus makes sense to describe the ideal of even integers as the smallest ideal that contains 2 : that is, every other ideal containing 2 contains the even integers, and the even integers are precisely the set of integers which are a multiple of 2 . These notions, of each even integer being a multiple of 2 , and of the even integers being the smallest such ideal containing 2 , motivates the notion of generators of an ideal. Definition 1.1.4. We say that an ideal I is generated by a subset of elements S ⊂ I if n P every element i ∈ I can be written in the form i = r s , for some positive integer n, and j j j =1 elements s , s , . . . , s ∈ S and r , r , . . . , r ∈ R. 1 2 n 1 2 n Similarly, given a ring R and elements s , s , . . . , s , we let ⟨ s , s , . . . , s ⟩ be the set of 1 2 n 1 2 n n P elements of the form i = r s for elements r , r , . . . , r ∈ R. We can also substitute a j j 1 2 n j =1 n P set, letting ⟨ S ⟩ be the set of elements in R of the form i = r s for some positive integer j j j =1 n, and elements s , s , . . . , s ∈ S, r , r , . . . , r ∈ R. 1 2 n 1 2 n With the notation above, the set of even integers, 2 Z , can also be written as ⟨ 2 ⟩ . Note that the set of even integers is an ideal. This happens more generally, as follows. Proposition 1.1.5. Given a subset S ⊂ R, the set ⟨ S ⟩ is an ideal. ′ Proof. For the first condition, suppose that we have two elements in ⟨ S ⟩ , say i and i . ′ ′ ′ Then, there are positive integers n, m and elements s , s , . . . , s , s , s , . . . , s ∈ S and 1 2 n 1 2 m n m P P ′ ′ ′ ′ ′ ′ r , r , . . . , r , r , r , . . . , r ∈ S such that i = r s and i = r s . Then, their sum 1 2 n j j m 1 2 j j j =1 j =1 n m P P ′ ′ ′ is equal to i + i = r s + r s , which is of the given form. Notice that we can j j j j j =1 j =1 ′ further simplify this expression if we know that some of the s and s are equal, using the j j distributive property. n P For the second condition, given an element i in ⟨ S ⟩ , we can write it as r s for some j j j =1 positive integer n, s , s , . . . , s ∈ S and r , r , . . . , r ∈ R. But then, for each element 1 2 n 1 2 n n P r ∈ R, observe that ri = rr s , which is also in ⟨ S ⟩ . This finishes the proof of the j j j =1 proposition. Definition 1.1.6. We call the set ⟨ S ⟩ the ideal generated by S. We now wish to generalize the notion of a prime from Z . Rather than thinking about elements as being primes, we want to think about ideals. The main behavior we want to capture is the fact that, given a prime number p, if ab lies in p Z , then either a or b lies in it. Contrast this with 6 Z , for instance: 2 · 3 lies in 6 Z , but 2 , 3 do not lie in 6 Z . Definition 1.1.7. An ideal I of a ring R is said to be prime if it is a proper ideal, and furthermore, for all a, b ∈ R, ab ∈ I implies that either a ∈ I or b ∈ I. For instance, the ideal ⟨ 2 ⟩ ⊂ Z is a prime ideal, since ab ∈ ⟨ 2 ⟩ if and only if ab is an even integer; but notice that one of a, b must be even as well.