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Commutative Number Integer Ring Subring Integral Closure Number Field Bibliography For Topology Subalgebra Of An Algebraic System Algebras Imaginary Quadratic Field Arithmetic Of Elliptic Curves

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Order In An Algebra

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

Examples:

The ring of integers in a number field is an order, known as the maximal order.
Let 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 (in fact it can be proved that every order of is of this form). The number is called the conductor of the order $\mathcal{O}$ .

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

1	Integer
1	Number
1	Ring
1	Subring
1	Integral Closure
1	Commutative
2	Bibliography For Topology
2	Subalgebra Of An Algebraic System
2	Number Field
3	Algebras
5	Imaginary Quadratic Field
15	Arithmetic Of Elliptic Curves