This chapter presents the basic commutative algebra tools for algebraic number theory. It begins with the Chinese Remainder Theorem for arbitrary commutative rings, which decomposes ring elements modulo the intersection of pairwise coprime ideals – illustrating a fundamental local – global principle. This principle is further refined through localization, a technique that isolates the behavior of rings at prime ideals that do not intersect a specified multiplicative subset.

A central class of rings in this study is Noetherian rings, characterized by the ascending chain condition on ideals, which ensures desirable finiteness properties. The Hilbert Basis Theorem guarantees that polynomial rings over Noetherian rings remain Noetherian, extending these finiteness conditions to broader algebraic contexts.

Further structural insights arise from studying fractional ideals and the Picard group, which classify invertible ideals and play a crucial role in the study of rings of Krull dimension 1, particularly Dedekind domains.

The chapter concludes with finite-dimensional commutative algebras over fields, a particular case of rings of Krull dimension 0. Under semisimple and separable conditions, these algebras decompose into direct products of fields, permitting a complete classification.