Peirce decomposition
Let be an idempotent of a ring , not necessarily with an identity. For any subset of , we introduce the notations:
and
If it happens that has an identity element, then is a legitimate element of , and this notation agrees with the usual product of an element and a set.
It is easy to see that for any set which contains .
Applying this first on the right with and then on the left with and , we obtain:
This is called the Peirce Decompostion of with respect to .
This is an example of a generalized matrix ring:
More generally, if has an identity element, and is a complete set of orthogonal idempotents, then
is a generalized matrix ring.
Title | Peirce decomposition |
---|---|
Canonical name | PeirceDecomposition |
Date of creation | 2013-03-22 14:39:17 |
Last modified on | 2013-03-22 14:39:17 |
Owner | mclase (549) |
Last modified by | mclase (549) |
Numerical id | 7 |
Author | mclase (549) |
Entry type | Definition |
Classification | msc 16S99 |