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.