generalized matrix ring

Let $I$ be an indexing set. A ring of $I\mathrm{\times }I$ generalized matricesis a ring $R$ with a decompostion (as an additive group)

$$R=\underset{i,j\in I}{\oplus }{R}_{ij},$$

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

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 $i\le i,j\le n$.

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

and we write elements of $R$ in the same fashion, then the multiplication in $R$ followsthe 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 $i\ne j$ an $R_i$-$R_j$-bimodule $R_{ij}$,and for each $i,j,k$ with $i\ne j$ and $j\ne k$a homomorphism $R_{ij}{\otimes }_{R_j}{R}_{jk}\to R_{ik}$,we can construct a generalized matrix ring structure on

$$R=\underset{i,j}{\oplus }{R}_{ij},$$

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