generalized matrix ring

Let I be an indexing set. A ring of I×I generalized matrices is a ring R with a decompostion (as an additive groupMathworldPlanetmath)


such that RijRklRil if j=k and RijRkl=0 if jk.

If I is finite, then we usually replace it by its cardinal n and speak of a ring of n×n generalized matrices with componentsMathworldPlanetmathPlanetmathPlanetmath Rij for ii,jn.

If we arrange the components Rij as follows:


and we write elements of R in the same fashion, then the multiplication in R follows the same pattern as ordinary matrix multiplicationMathworldPlanetmath.

Note that Rij is an Rii-Rjj-bimodule, and the multiplication of elements induces homomorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath RijRjjRjkRik for all i,j,k.

Conversely, given a collectionMathworldPlanetmath of rings Ri, and for each ij an Ri-Rj-bimodule Rij, and for each i,j,k with ij and jk a homomorphism RijRjRjkRik, we can construct a generalized matrix ring structureMathworldPlanetmath on


where we take Rii=Ri.

Title generalized matrix ring
Canonical name GeneralizedMatrixRing
Date of creation 2013-03-22 14:39:14
Last modified on 2013-03-22 14:39:14
Owner mclase (549)
Last modified by mclase (549)
Numerical id 4
Author mclase (549)
Entry type Definition
Classification msc 16S50