generalized matrix ring
Let be an indexing set. A ring of generalized matrices
is a ring with a decompostion (as an additive group)
such that if and if .
If is finite, then we usually replace it by its cardinal
and speak of a ring of generalized matrices with components
for .
If we arrange the components as follows:
and we write elements of in the same fashion, then the multiplication in follows
the same pattern as ordinary matrix multiplication.
Note that is an --bimodule,
and the multiplication of elements induces homomorphisms
for all .
Conversely, given a collection of rings ,
and for each an --bimodule ,
and for each with and
a homomorphism ,
we can construct a generalized matrix ring structure
on
where we take .
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 |