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...
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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...
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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...
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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)...
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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...
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Commutative ring (section Principal ideal domains)
important facts in commutative algebra, the going-up theorem and Krull's principal ideal theorem. A ring homomorphism or, more colloquially, simply a...
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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...
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Krull rings. To this day, Krull's principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra. These...
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sequence (algebra) Krull dimension Krull's principal ideal theorem Primary ideal Primary decomposition and the Lasker–Noether theorem Noether normalization...
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algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated...
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mathematical concepts: Krull dimension Krull's principal ideal theorem Krull's theorem Krull–Akizuki theorem Krull–Schmidt theorem Krull topology, an example...
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Noetherian ring (section Key theorems)
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...
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Kronecker–Weber theorem (number theory) Krull's principal ideal theorem (commutative algebra) Krull–Schmidt theorem (group theory) Kruskal's tree theorem (order...
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integer Theorems and applications Algebraic geometry Hilbert's Nullstellensatz Hilbert's basis theorem Hopkins–Levitzki theorem Krull's principal ideal theorem...
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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...
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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"...
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topology Krull–Azumaya theorem Krull–Schmidt category Krull–Schmidt theorem Krull's intersection theorem Krull's principal ideal theorem Krull's separation...
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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...
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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...
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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...
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Local ring (redirect from Krull intersection theorem)
_{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...
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denominators in I {\displaystyle I} , hence the name fractional ideal. The principal fractional ideals are those R {\displaystyle R} -submodules of K {\displaystyle...
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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...
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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...
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Primary decomposition (redirect from Lasker-Noether theorem)
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...
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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...
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Serre's criterion for normality (category Theorems in ring theory)
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...
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are UFDs: All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see Fundamental theorem of arithmetic)...
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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...
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Glossary of commutative algebra (redirect from Saturated ideal)
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...
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