In 1877 the author published an introduction to his ideal theory first in Bull. Sci. Math. and then in a book [Sur la théorie des nombres entiers algébriques, Gauthier-Villars, Paris (1877; JFM 09.0126.01)]. A German translation appeared in 1964 (Zbl 0124.02205), and here is the first English translation of this treatise, accompanied by a historical introduction of John Stillwell, the translator.
Dedekind introduces his theory in a leisurely fashion, starting with main properties of finitely generated subgroups (called here modules) of the additive group of complex numbers, then considering the rings \(\mathbb{Z}\), \(\mathbb{Z}[i]\) and \(\mathbb{Z}[\sqrt{-5}]\), and finally developing the theory of ideals in rings of integers of any algebraic number field of finite degree. It is remarkable that this classical text did not age and it may be still used in courses introducing classical ideal theory. Reading it, one sees how much most later authors of books dealing with this theory were obliged to Dedekind.