principal ideal domain
11Ideal norm — In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an… …
12Dedekind domain — In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily …
13Bézout domain — In mathematics, a Bézout domain is an integral domain which is, in a certain sense, a non Noetherian analogue of a principal ideal domain. More precisely, a Bézout domain is a domain in which every finitely generated ideal is principal. A… …
14Unique factorization domain — In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements, analogous to the fundamental theorem of… …
15Ascending chain condition on principal ideals — In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two sided ideals of a ring, partially ordered by inclusion. The ascending ascending chain condition on principal… …
16Euclidean domain — In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies. A Euclidean domain is a specific type of integral domain, and can be characterized by the following (not… …
17Integrally closed domain — In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in the field of fractions of A is A itself. Many well studied domains are integrally closed: Fields, the ring of integers Z, unique factorization… …
18Prime ideal — In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on… …
19Maximal ideal — In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals.[1][2] In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and… …
20Norm of an ideal — The norm of an ideal is defined in algebraic number theory. Let Ksubset L be two number fields with rings of integers O Ksubset O L. Suppose that the extension L/K is a Galois extension with :G= extstyle{Gal}(L/K). The norm of an ideal I of O L… …
