In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results.
The double centralizer theorems give conditions under which one can conclude that equality occurs.
Perhaps the most common version is the version for central simple algebras, as it appears in (Knapp 2007, p.115): Theorem: If A is a finite-dimensional central simple algebra over a field F and B is a simple subalgebra of A, then CA(CA(B)) = B, and moreover the dimensions satisfy The following generalized version for Artinian rings (which include finite-dimensional algebras) appears in (Isaacs 2009, p.187).
Given a simple R module UR, we will borrow notation from the above motivation section including RU and E=End(U).
Theorem: Let R be a right Artinian ring with a simple right module UR, and let RU, D and E be given as in the previous paragraph.