Commutative Rings

Commutative ring theory has its origins in number theory and algebraic geometry in the 19th century. Today it is of particular importance in algebraic geometry, and there has been an interesting interaction of algebraic geometry and number theory, using the methods of commutative algebra. Here we can do no more than describe the basic techniques and take the first steps in the subject. In Section 10.1 we define the various operations on ideals and use them in Section 10.2 to study unique factorization . In Section 10.3 we give an account of fractions and examine the effect of chain conditions in Section lOA. Many rings of algebraic numbers fail to have unique factorization of elements, but instead have unique factorization of ideals, and the consequences are studied in Sections 10.5 and 10.6. Sections 10.7–10.10 deal with the properties of rings used in algebraic geometry (but also of importance in commutative ring theory): equations (Section 10.7), decomposition of ideals (Section 10.8), dimension (Section 10.9) and the relation between ideals and algebraic varieties (Section 10.10).