By N. Saavedra Rivano

Publication by means of Saavedra Rivano, N.

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On the other hand, if a ∈ R and a ∈ / N , then we must show that there is a prime ideal P such that a ∈ / P , and hence that a is not an element of the intersection of all prime ideals of R. In order to do this we need to use Zorn’s lemma. As is always the case when using Zorn’s lemma, the key is a careful selection of the set S. In this case, we want a maximal element of S to 48 Chapter 5 be a prime ideal P such that a ∈ / P . With this in mind, we let S be the / I for all positive integers n — that set of all ideals I of R such that an ∈ is, neither a nor any of its powers an lie in I .

The ring Z/(5) is often called the ring of integers modulo 5, and is sometimes written as Z5 . Ideals and Quotient Rings 19 The term “quotient ring” and the notation “R/I ” can be somewhat misleading since they have nothing to do with quotients or fractions of numbers. Rather, both the notation and the term are meant to suggest that the ideal I factors or partitions or divides the ring R into cosets, each having the same size as I . However, as long as you read R/I as “R mod I ”, and remember that the elements of this ring are cosets, there is little danger of confusing the concept of a quotient ring with the still-to-come notion of a ring of fractions.

In fact, we need to be very careful not to assume countability in a proof using chains. We are getting closer to Zorn’s lemma, but we still need a few more concepts. 3. Let ≤ be a partial order on a set S. Let A be a subset of S. An upper bound of the set A is an element s ∈ S such that a ≤ s, for all a ∈ A. Zorn’s Lemma and Maximal Ideals 39 An upper bound of a set A is simply an element that is bigger than (or possibly equal to) every element in the set A. A set A might have many upper bounds, or a single upper bound, or even possibly no upper bounds at all.