algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian...

In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single...

In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the...

element of the ring. By the principal ideal theorem, any non-principal ideal becomes principal when extended to an ideal of the Hilbert class field. This...

a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is...

classification of principal rings in terms of special principal rings and principal ideal domains. Zariski–Samuel theorem: Let R be a principal ring. Then R...

algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated...

ideal theorem use minimal prime ideals. A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing...

concepts: Krull dimension Krull's principal ideal theorem Krull's theorem Krull–Akizuki theorem Krull–Schmidt theorem Krull topology, an example of the...

function Homological conjectures in commutative algebra Krull's principal ideal theorem Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989...

(algebra) Krull dimension Krull's principal ideal theorem Primary ideal Primary decomposition and the Lasker–Noether theorem Noether normalization lemma Going...

algebra) Integral root theorem (algebra, polynomials) Krull's principal ideal theorem (commutative algebra) Lasker–Noether theorem (commutative algebra)...

non-Noetherian local ring whose maximal ideal is principal (see a counterexample to Krull's intersection theorem at Local ring#Commutative case.) If R is...

generators of m {\displaystyle {\mathfrak {m}}} . Then Krull's principal ideal theorem implies that n ≥ dim ⁡ A {\displaystyle n\geq \dim A} , and A {\displaystyle...

important facts in commutative algebra, the going-up theorem and Krull's principal ideal theorem. A ring homomorphism or, more colloquially, simply a...

mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for...

and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when...

ideals cannot be derived within ZF (Zermelo–Fraenkel set theory without the axiom of choice). This issue is discussed in various prime ideal theorems...

principal ideal theorem, every prime ideal of O generates a principal ideal of the ring of integers of E. A generator of this principal ideal is called...

long history in commutative algebra. Krull’s famous proof of his principal ideal theorem uses them in an essential way. They first arose after primary decompositions...

denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...

descendants. He is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper...

small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any...

ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can...

rings. To this day, Krull's principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra. These...

principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals...

modern algebraic geometry. The two other theorems are Hilbert's basis theorem, which asserts that all ideals of polynomial rings over a field are finitely...

number fields with class number one Principal ideal domain Algebraic K-theory Galois theory Fermat's Last Theorem Narrow class group Picard group—a generalisation...

contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull's theorem. More generally, if S is...

proving the principal ideal theorem in class field theory. See the Emil Artin-John Tate Class Field Theory notes. Focal subgroup theorem, an important...