Amplitude amplification

In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization. They were introduced by Template:Harvs. They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

In this article, a ring is commutative and has unity.

Formal definition

Let $A$ be an integral domain and let $P$ be the set of all prime ideals of $A$ of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then $A$ is a Krull ring if and only if

1. $A_{\mathfrak {p}}$ is a discrete valuation ring for all ${\mathfrak {p}}\in P$,
2. $A$ is the intersection of these discrete valuation rings (considered as subrings of the quotient field of $A$).
3. Any nonzero element of $A$ is contained in only a finite number of height 1 prime ideals.

Properties

A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.

Let A be a Zariski ring (e.g., a local noetherian ring). If the completion ${\widehat {A}}$ is a Krull domain, then A is a Krull domain.

Examples

1. Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
2. If $A$ is a Krull ring then so is the polynomial ring $A[x]$ and the formal power series ring $A[[x]]$.
3. The polynomial ring $R[x_{1},x_{2},x_{3},\ldots ]$ in infinitely many variables over a unique factorization domain $R$ is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
4. Let $A$ be a Noetherian domain with quotient field $K$, and $L$ be a finite algebraic extension of $K$. Then the integral closure of $A$ in $L$ is a Krull ring (Mori–Nagata theorem).

The divisor class group of a Krull ring

A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form div(x) for some non-zero x in A is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A.

A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A).

Example: in the ring k[x,y,z]/(xy–z²] the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.

References

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• Hideyuki Matsumura, Commutative Algebra. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
• Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN 0-521-25916-9
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
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