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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, ring, integer, quadratic imaginary field, number field, ring of integers, subring, finitely generated, commutative, algebra
There are 38 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 6087 times total.

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