Given a field or a commutative algebra k, new algebras can be built
from a finite number of matrix algebras over k (of varying size) by
slightly changing the multiplication. This provides a structure that
occurs frequently; it covers, for instance, group algebras of symmetric
groups and various Hecke algebras. This structure can be used for
classifying and, to some extent, describing simple modules. Moreover,
and perhaps unexpectedly, some homological properties have been found,
too.