The Subfield Generated by a Subring and Localization (Algebra 2: Lecture 1 Video 4)

Lecture 1: In this first lecture we discussed Rings of Fractions following the presentation in Section 7.5 of Dummit and Foote.  We first discussed the rational numbers and how we can think of the elements as equivalence classes of ordered pairs of integers.  We then stated the main theorem of this lecture, Theorem 15 in Section 7.5 of Dummit and Foote.  We gave several examples of the Field of Fractions of an integral domain.  For example, we discussed the field of fractions of the polynomial ring R[x].  We then proved the main theorem and also proved a corollary about the subfield generated a subring of a field.  At the end of the lecture we discussed a generalization of the main result we proved where we now allow the set D to contain zero divisors.  We saw one important example, the localization of a ring at a prime ideal.

Reading: In this lecture we closely followed the presentation in Section 7.5.  You should read the entire section.  We also stated Theorem 36 and Corollary 37 right after it in Section 15.4.  The proof of this theorem is very similar to our proof from lecture, and you should read it.  We briefly discussed Example 3 on page 708.