ideal of an algebra


left idealMathworldPlanetmathPlanetmath \PMlinkescapephraseright ideal \PMlinkescapephrasetwo-sided ideal

Let A be an algebra over a ring R.

Definition - A left ideal of A is a subalgebraMathworldPlanetmath IA such that axI whenever aA and xI.

Equivalently, a left ideal of A is a subset IA such that

  1. 1.

    x-yI, for all x,yI.

  2. 2.

    rxI, for all rR and xI.

  3. 3.

    axI, for all aA and xI

Similarly one can define a right ideal by replacing condition 3 by: xaI whenever aA and xI.

A two-sided ideal of A is a left ideal which is also a right ideal. Usually the word ”” by itself means two-sided ideal. Of course, all these notions coincide when A is commutativePlanetmathPlanetmathPlanetmath.

0.0.1 Remark

Since an algebra is also a ring, one might think of borrowing the definition of ideal from ring . The problem is that condition 2 would not be in general satisfied (unless the algebra is unital).

Title ideal of an algebra
Canonical name IdealOfAnAlgebra
Date of creation 2013-03-22 18:09:00
Last modified on 2013-03-22 18:09:00
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Definition
Classification msc 16D25
Synonym left ideal of an algebra
Synonym right ideal of an algebra
Synonym two-sided ideal of an algebra