Dedekind domains

A Dedekind domain is a type of mathematical structure in algebra, specifically a ring, which allows for unique factorization of elements into prime elements, similar to integers factoring into prime numbers. Unlike some rings, where elements can factor in complicated ways, Dedekind domains ensure that every non-zero ideal (a kind of subset) can be uniquely broken down into prime ideal factors. This property makes Dedekind domains a powerful framework for number theory and algebraic geometry, providing a well-behaved environment for studying algebraic structures and their properties, especially in contexts involving algebraic integers and number fields.