ideaalien

Ideaalien is the Dutch plural form used for the mathematical concept of ideals, particularly in ring theory. In modern Dutch texts the standard plural is usually idealen, while ideaalien may appear as an alternate form in some sources. In English-language mathematics, the corresponding concept is called an ideal (plural ideals).

Definition and basic idea. In a ring R, an ideal I is a subset that is an additive subgroup of R and is closed under multiplication by any ring element. More precisely, for every a in R and x in I, both ax and xa belong to I. In noncommutative rings, one distinguishes left ideals (closed under ax) and right ideals (closed under xa); a two-sided ideal is closed under both.

Examples. In the ring of integers Z, the set nZ for any n ≥ 0 is an ideal. In the polynomial ring k[x], the principal ideal (f) consists of all multiples f g with g in k[x]. The zero ideal {0} and the whole ring R are the two trivial ideals. In a commutative ring, all ideals are two-sided by default.

Key properties and constructions. The sum and intersection of ideals are ideals, as is the product of ideals. For an ideal I in R, one forms the quotient ring R/I, which encodes the structure of R modulo I. Maximal ideals I have the property that R/I is a field; prime ideals I yield R/I as an integral domain. Ideals also underpin notions such as radical, primary decomposition, and various decomposition theorems in Noetherian rings.

Usage and significance. Ideals provide a framework for studying decompositions, quotient structures, and homomorphic images in algebra. They play a central role across many areas of mathematics, including number theory, algebraic geometry, and algebraic topology.