Idealerne

Idealerne, or the ideals, are a fundamental concept in ring theory, a branch of abstract algebra. If R is a ring, an ideal I ⊆ R is an additive subgroup that is closed under multiplication by elements of R: for every r in R and i in I, both ri and ir belong to I. In a commutative ring, left and right ideals coincide and are simply called ideals. An ideal generated by a subset S ⊆ R is the smallest ideal containing S, commonly denoted (S). A principal ideal is one generated by a single element, written (a).

Idealerne can be ordered by inclusion and form a lattice under this relation. They provide a way to construct new rings via quotients: the quotient ring R/I consists of cosets of I, with well-defined addition and multiplication. Quotients relate to homomorphisms through the First Isomorphism Theorem, and many structural results are expressed via ideal correspondences, such as the correspondence between subgroups of a domain and its quotient structures.

Prime and maximal ideals are central classes. A proper ideal P is prime if whenever ab ∈ P, then a ∈ P or b ∈ P. A maximal ideal M is such that R/M is a field, equivalently there is no larger proper ideal than M. The set of prime ideals, Spec(R), is equipped with a topology (the Zariski topology) and serves as a bridge between algebra and geometry in algebraic geometry.

Examples illustrate the concept. In the ring of integers Z, every ideal is of the form nZ. In a principal ideal domain like Z or k[x], every ideal is principal, while in multivariate polynomial rings such as k[x,y], not all ideals are principal. Idealerne thus anchor many constructions, from factorization and quotient structures to geometric interpretations.