Über die Theorie der ganzen algebraischen Zahlen

Richard Dedekind's work, Über die Theorie der ganzen algebraischen Zahlen (1879), introduced the concept of an "ideal" in mathematics. An ideal is a specific type of set of numbers, defined by properties derived from ordinary arithmetic.12

Dedekind's theory focused on algebraic integers, which are solutions to polynomial equations with integer coefficients where the leading coefficient is 1. He defined an ideal as a collection of algebraic integers that could be separated from a larger set and satisfied two key properties: the difference between any two numbers in the set is also in the set, and the product of any number in the set with any integer is also in the set.32

This concept of ideals allowed Dedekind to generalize the idea of unique factorization, which is fundamental to number theory, to various algebraic structures that were previously difficult to analyze. His work significantly influenced modern algebra by emphasizing set-theoretical approaches and structural properties.123