In commutative 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 ring. The theorem is sometimes referred to by its German name, Krulls...
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 element ...
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the...
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal prime ideals.
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely...
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said to be left-Noetherian or...
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not...
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the...
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of...