Peirce decomposition

Let e be an idempotentPlanetmathPlanetmath of a ring R, not necessarily with an identityPlanetmathPlanetmath. For any subset X of R, we introduce the notations:




If it happens that R has an identity elementMathworldPlanetmath, then 1-e is a legitimate element of R, and this notation agrees with the usual product of an element and a set.

It is easy to see that XeX(1-e)=0=eX(1-e)X for any set X which contains 0.

Applying this first on the right with X=R and then on the left with X=Re and X=R(1-e), we obtain:


This is called the Peirce Decompostion of R with respect to e.

Note that eRe and (1-e)R(1-e) are subrings, eR(1-e) is an eRe-(1-e)R(1-e)-bimodule, and (1-e)Re is a (1-e)R(1-e)-eRe-bimodule.

This is an example of a generalized matrix ring:


More generally, if R has an identity element, and e1,e2,,en is a complete set of orthogonal idempotents, then


is a generalized matrix ring.

Title Peirce decompositionMathworldPlanetmath
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