# generalized matrix ring

Let $I$ be an indexing set. A ring of $I\mathrm{\times}I$ generalized matrices
is 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:

$$\left(\begin{array}{cccc}\hfill {R}_{11}\hfill & \hfill {R}_{12}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {R}_{1n}\hfill \\ \hfill {R}_{21}\hfill & \hfill {R}_{22}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {R}_{2n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {R}_{n1}\hfill & \hfill {R}_{n2}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {R}_{nn}\hfill \end{array}\right)$$ |

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 $i\ne j$ 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 ${R}_{ii}={R}_{i}$.

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 |