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order in an algebra (Definition)

Let $ A$ be an algebra (not necessarily commutative), finitely generated over $ \mathbb{Q}$. An order $ R$ of $ A$ is a subring of $ A$ which is finitely generated as a $ \mathbb{Z}$-module and which satisfies $ R \otimes \mathbb{Q}= A$.

Examples:

  1. The ring of integers in a number field is an order, known as the maximal order.
  2. Let $ K$ be a quadratic imaginary field and $ \mathcal{O}_K$ its ring of integers. For each integer $ n\geq 1$ the ring $ \mathcal{O}={\mathbb{Z}}+n\mathcal{O}_K$ is an order of $ K$ (in fact it can be proved that every order of $ K$ is of this form). The number $ n$ is called the conductor of the order $ \mathcal{O}$.

Reference: Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1986.



"order in an algebra" is owned by alozano.
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See Also: complex multiplication

Also defines:  order, maximal order, conductor of an order
Keywords:  order, maximal order, algebra, ring of integers

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orders in a number field (Topic) by pahio
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Cross-references: the arithmetic of elliptic curves, number, ring, integer, quadratic imaginary field, number field, ring of integers, subring, finitely generated, commutative, algebra
There are 37 references to this entry.

This is version 7 of order in an algebra, born on 2003-06-14, modified 2007-05-15.
Object id is 4362, canonical name is OrderInAnAlgebra.
Accessed 6312 times total.

Classification:
AMS MSC: 06B10 (Order, lattices, ordered algebraic structures :: Lattices :: Ideals, congruence relations)
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