Principal ideal theorem information

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...

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, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element...

nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This...

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...

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

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...

..., an is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A. The appellation...

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

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

element theorem (field theory) Principal axis theorem (linear algebra) Principal ideal theorem (algebraic number theory) Prokhorov's theorem (measure...

fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization...

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...

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...

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...

function Homological conjectures in commutative algebra Krull's principal ideal theorem Regular local ring Matsumura, Hideyuki: "Commutative Ring Theory"...

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

integer Theorems and applications Algebraic geometry Hilbert's Nullstellensatz Hilbert's basis theorem Hopkins–Levitzki theorem Krull's principal ideal theorem...

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

ideals of R. Krull's theorem (1929): Every nonzero unital ring has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal"...

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...

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

topology Krull–Azumaya theorem Krull–Schmidt category Krull–Schmidt theorem Krull's intersection theorem Krull's principal ideal theorem Krull's separation...