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integral (Definition)

Let $ B$ be a ring with a subring $ A$. We will assume that $ A$ is contained in the center of $ B$ (in particular, $ A$ is commutative). An element $ x \in B$ is integral over $ A$ if there exist elements $ a_0, \dots, a_{n-1} \in A$ such that

$\displaystyle x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0. $
The ring $ B$ is integral over $ A$ if every element of $ B$ is integral over $ A$.



"integral" is owned by djao.
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See Also: integral basis


Attachments:
integral element (Definition) by pahio
characterizations of integral (Theorem) by pahio
integrality is transitive (Theorem) by rm50
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Cross-references: commutative, center, contained, subring, ring
There are 93 references to this entry.

This is version 5 of integral, born on 2002-01-05, modified 2005-02-14.
Object id is 1295, canonical name is Integral.
Accessed 9081 times total.

Classification:
AMS MSC: 13B21 (Commutative rings and algebras :: Ring extensions and related topics :: Integral dependence)
Pending Errata and Addenda
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example / definition by silence on 2003-09-15 10:59:13
see for example...
http://mathworld.wolfram.com/CorrelationIntegral.html

for a definition of the correlation integral.
[ reply | up ]
correlation integral by silence on 2003-09-15 10:57:10
There is a lot of literature pertaining to correlation sums, correlation integrals, and so on. I would like to know if anyone can give me a concise intuitive way of thinking about how to interpret the correlation integral? Specifically, what property of a time-series is captured in its correlation integral? I would appreciate any help.
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