generalized matrix ring

Let $I$ be an indexing set. A ring of $I\times I$ generalized matrices is a ring $R$ with a decompostion (as an additive group)

R=\bigoplus_{i,j\in I}R_{ij},

such that $R_{ij}R_{kl}\subseteq R_{il}$ if $j=k$ and $R_{ij}R_{kl}=0$ if $j\neq k$ .

If $I$ is finite, then we usually replace it by its cardinal $n$ and speak of a ring of $n\times n$ generalized matrices with components $R_{ij}$ for $i\leq i,j\leq n$ .

If we arrange the components $R_{ij}$ as follows:

\begin{pmatrix}R_{11}&R_{12}&\dots&R_{1n}\\ R_{21}&R_{22}&\dots&R_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ R_{n1}&R_{n2}&\dots&R_{nn}\end{pmatrix}

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

Note that $R_{ij}$ is an $R_{ii}$ - $R_{jj}$ -bimodule, and the multiplication of elements induces homomorphisms $R_{ij}\otimes_{R_{jj}}R_{jk}\to R_{ik}$ for all $i, j, k$ .

Conversely, given a collection of rings $R_{i}$ , and for each $i\neq j$ an $R_{i}$ - $R_{j}$ -bimodule $R_{ij}$ , and for each $i, j, k$ with $i\neq j$ and $j\neq k$ a homomorphism $R_{ij}\otimes_{R_{j}}R_{jk}\to R_{ik}$ , we can construct a generalized matrix ring structure on

R=\bigoplus_{i,j}R_{ij},

where we take $R_{ii}=R_{i}$ .

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