ideal generated by a subset of a ring
Let be a subset of a ring . Let be the collection of all left ideals of that contain (note that the set is nonempty since and is an ideal in itself). The intersection
is called the left ideal generated by , and is denoted by . We say that generates 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 consists of finite -linear combinations of elements of :
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 |