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...

R.} Krull's principal ideal theorem states that if R {\displaystyle R} is a Noetherian ring and I {\displaystyle I} is a principal, proper ideal of R...

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...

a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's height theorem and its converse)...

suppose a1, ..., an is a minimal set of generators of m. Then Krull's principal ideal theorem implies that n ≥ dim A, and A is regular whenever n = dim A...

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

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...

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

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

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

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

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...

Kronecker–Weber theorem (number theory) Krull's principal ideal theorem (commutative algebra) Krull–Schmidt theorem (group theory) Kruskal's tree theorem (order...

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

ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see Krull's theorem for more. An arbitrary union of ideals...

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"...

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

fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal p {\displaystyle...

that involves, in particular, Hilbert's Nullstellensatz and Krull's principal ideal theorem. A system is zero-dimensional if it has a finite number of...

p 1 {\displaystyle {\mathfrak {p}}_{1}} has height one by Krull's principal ideal theorem and p 2 {\displaystyle {\mathfrak {p}}_{2}} has height two...

_{i=1}^{\infty }m^{i}=\{0\}} (Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. The theorem is a consequence of...

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

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

1 prime ideals. It is also possible to characterize Krull rings by mean of valuations only: An integral domain A {\displaystyle A} is a Krull ring if...

mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection...

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...

has height one (Krull's principal ideal theorem). For R1, we argue in the same way: let p {\displaystyle {\mathfrak {p}}} be a prime ideal of height one...

are UFDs: All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see Fundamental theorem of arithmetic)...

well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding...

prime ideals in R and q2 is a prime ideal of S with q2∩R=p2, there is a prime ideal q1 of S with q1⊆q2 and q1∩R=p1. 2.  The going down theorem states...