characterizations of integral


Let R be a subring of a field K,  1R  and let α be a non-zero element of K.  The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    α is integral over R.

  2. 2.

    α belongs to R[α-1].

  3. 3.

    α is unit of R[α-1].

  4. 4.


Proof.  Supposing the first condition that an equation


with aj’s belonging to R, holds.  Dividing both by αn-1 gives


One sees that α belongs to the ring R[α-1] even being a unit of this (of course  α-1R[α-1]).  Therefore also the principal idealPlanetmathPlanetmathPlanetmathPlanetmath α-1R[α-1] of the ring R[α-1] coincides with this ring.  Conversely, the last circumstance implies that α is integral over R.


  • 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title characterizations of integral
Canonical name CharacterizationsOfIntegral
Date of creation 2013-03-22 14:56:54
Last modified on 2013-03-22 14:56:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 12E99
Classification msc 13B21
Synonym characterisations of integral