order in an algebra

Let $A$ be an algebra (not necessarily commutative), finitely generated over $\\mathbb{Q}$ . An order $R$ of $A$ is a subringof $A$ 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 asthe maximal order.
2.
Let $K$ be a quadratic imaginary field and $\\mathcal{O}_{K}$ itsring 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 beproved that every order of $K$ is of this form). The number $n$ is called the of the order $\\mathcal{O}$ .

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

单词	OrderInAnAlgebra
释义	order in an algebra Let $A$ be an algebra (not necessarily commutative), finitely generated over $\\mathbb{Q}$ . An order $R$ of $A$ is a subringof $A$ 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 asthe maximal order. 2. Let $K$ be a quadratic imaginary field and $\\mathcal{O}_{K}$ itsring 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 beproved that every order of $K$ is of this form). The number $n$ is called the of the order $\\mathcal{O}$ . Reference: Joseph H. Silverman, The arithmetic ofelliptic curves, Springer-Verlag, New York, 1986.
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