Dedekindide

Dedekindide is a term used in the field of mathematics, specifically in the context of Dedekind domains and Dedekind rings. A Dedekind domain is an integral domain in which every non-zero ideal factors uniquely into a product of prime ideals. This property is analogous to the unique factorization of integers into prime numbers. Dedekind domains are named after the German mathematician Richard Dedekind, who introduced the concept in the late 19th century.

Dedekind domains are a generalization of principal ideal domains (PIDs), which are integral domains where every ideal is principal. In a Dedekind domain, not every ideal is necessarily principal, but every non-zero ideal can be expressed as a product of prime ideals in a unique way. This unique factorization property is crucial in many areas of mathematics, including algebraic number theory and algebraic geometry.

Dedekind rings are a further generalization of Dedekind domains, allowing for the inclusion of zero divisors. In a Dedekind ring, every non-zero ideal still factors uniquely into a product of prime ideals, but the ring itself may contain elements that are not regular (i.e., elements that are zero divisors).

The study of Dedekind domains and Dedekind rings is fundamental in the understanding of algebraic structures and their applications in various branches of mathematics. These concepts provide a framework for exploring the properties of ideals and their factorization, which is essential for deeper investigations into the arithmetic of rings and fields.