In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need not contain their own centralizers, which would be pretty bad.
In finite groups, subgroups that contain their own centralizers have an eery importance. It occurred to me that subalgebras of matrix algebras probably have some similar properties, but I don't know what to search for.
What do you call a subalgebra that contains its own centralizer?
In other words, is there a name for a collection of matrices:
- that contains the scalar matrices,
- that is closed under addition and multiplication, and
- that contains every matrix that commutes with every matrix in the collection?