Commutative Algebra

We will do some serious commutative algebra in this chapter, which will provide a powerful algebraic foundation for understanding the more refined number-theoretic structures associated to number fields.

In the first section we establish the standard properties of Noetherian rings and modules, including the Hilbert basis theorem. We also observe that finitely generated abelian groups are Noetherian $\mathbf{Z}$-modules, which fills the gap in our proof of the structure theorem for finitely generated abelian groups. After establishing properties of Noetherian rings, we consider the rings of algebraic integers and discuss some of their properties.