Korrektiivsete

Korrektiivsete, also known as corrective sets, are a type of mathematical object used in the study of algebraic structures, particularly in the context of group theory and ring theory. They were introduced by the mathematician Richard Dedekind in the late 19th century as a tool to understand the structure of ideals in rings.

A corrective set is a subset of a ring that satisfies certain properties, making it useful for studying the divisibility and factorization of elements within the ring. Specifically, a corrective set is a subset that is closed under multiplication and contains the identity element of the ring. This closure property allows for the construction of a new ring, known as the corrective ring, which can provide insights into the original ring's structure.

Corrective sets are particularly important in the study of Dedekind domains, which are a class of integral domains that generalize the properties of principal ideal domains. In Dedekind domains, every nonzero ideal can be factored uniquely into a product of prime ideals, a property known as unique factorization. Corrective sets play a role in understanding this factorization by providing a way to relate different ideals within the domain.

The concept of corrective sets has also been extended to more general algebraic structures, such as modules and lattices, where similar closure properties are used to study the structure and properties of these objects. Overall, corrective sets are a fundamental tool in algebraic geometry and number theory, providing a means to analyze the internal structure of rings and modules.