order in an algebra

Let A be an algebra (not necessarily commutativePlanetmathPlanetmathPlanetmath), finitely generatedMathworldPlanetmathPlanetmath over . An order R of A is a subring of A which is finitely generated as a -module and which satisfies R=A.


  1. 1.

    The ring of integersMathworldPlanetmath in a number field is an order, known as the maximal orderMathworldPlanetmath.

  2. 2.

    Let K be a quadratic imaginary field and 𝒪K its ring of integers. For each integer n1 the ring 𝒪=+n𝒪K 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 of the order 𝒪.

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

Title order in an algebra
Canonical name OrderInAnAlgebra
Date of creation 2013-03-22 13:41:22
Last modified on 2013-03-22 13:41:22
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Definition
Classification msc 06B10
Related topic ComplexMultiplication
Defines order
Defines maximal order
Defines conductorPlanetmathPlanetmathPlanetmath of an order