The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).

# REDIRECT Krull's principal ideal theorem

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 Noetherian ring.

This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.

# REDIRECT Krull's principal ideal theorem

** Krull's principal ideal theorem

* Krull's principal ideal theorem

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.

Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.

* Krull's theorem can fail for rings without unity.

** Krull's intersection theorem

** Krull's theorem

* Krull's intersection theorem

* Krull's theorem

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

Bézout's lemma is true in any principal ideal domain, but there are integral domains in which it is not true.

As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).

then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra + Rb is principal and indeed is equal to Rd.

Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the Fundamental Theorem of Arithmetic: every Euclidean domain is a unique factorization domain.

It is important to compare the class of Euclidean domains with the larger class of principal ideal domains ( PIDs ).

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

In words, one may define f ( a ) to be the minimum value attained by g on the set of all non-zero elements of the principal ideal generated by a.

The property ( EF1 ) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R / I have a representative r with.

* R is a principal ideal domain.

In modern mathematical language, the ideal generated by a and b is the ideal generated by g alone ( an ideal generated by a single element is called a principal ideal, and all ideals of the integers are principal ideals ).

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.

Important examples are Euclidean domains and principal ideal domains.

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

This is similar ( but more categorical ) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is " not natural ", or rather " not unique ", as automorphisms exist that do not preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain # Uniqueness for example.

Thus, by the principal axis theorem, the second fundamental form is

In view of the 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 theory for finitely generated modules over a Dedekind domain.

The same work contained the celebrated formula known as Taylor's theorem, the importance of which remained unrecognised until 1772, when J. L. Lagrange realized its powers and termed it " le principal fondement du calcul différentiel " (" the main foundation of differential calculus ").

* The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

The Quillen – Suslin theorem, which solves Serre's problem is another deep result ; it states that if K is a field, or more generally a principal ideal domain, and R = K is a polynomial ring over K, then every projective module over R is free.

The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting.

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

structure theorem for finitely generated modules over a principal ideal domain:

Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.

He is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper ( 1929 ).