ideal generated by a subset of a ring

Let X be a subset of a ring R. Let S={Ik} be the collection of all left idealsMathworldPlanetmath of R that contain X (note that the set is nonempty since XR and R is an ideal in itself). The intersection


is called the left ideal generated by X, and is denoted by (X). We say that X generates I as an ideal.

The definition is symmetrical for right ideals.

Alternatively, we can constructively form the set of elements that constitutes this ideal: The left ideal (X) consists of finite R-linear combinationsMathworldPlanetmath of elements of X:

Title ideal generated by a subset of a ring
Canonical name IdealGeneratedByASubsetOfARing
Date of creation 2013-03-22 14:39:04
Last modified on 2013-03-22 14:39:04
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 16D25
Related topic GeneratorsOfInverseIdeal
Related topic PrimeIdealsByKrullArePrimeIdeals
Defines ideal generated by
Defines left ideal generated by
Defines right ideal generated by
Defines generate as an ideal
Defines generates as an ideal
Defines generates