Basic Commutative Algebra

The commutative algebra in this chapter provides a foundation for understanding the more refined number-theoretic structures associated to number fields.

First we prove the structure theorem for finitely generated abelian groups. Then we establish the standard properties of Noetherian rings and modules, including a proof of the Hilbert basis theorem. We also observe that finitely generated abelian groups are Noetherian $\mathbf {Z}$-modules. After establishing properties of Noetherian rings, we consider rings of algebraic integers and discuss some of their properties.