# 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 GeneralizedMatrixRing 2013-03-22 14:39:14 2013-03-22 14:39:14 mclase (549) mclase (549) 4 mclase (549) Definition msc 16S50