Balanced module

In the subfield of abstract algebra known as module theory, a right "R" module "M" is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group "M" which commutes with all "R"-endomorphisms of "M" is given by multiplication by a ring element. Explicitly, for any additive endomorphism "f", if "fg" = "gf" for every "R" endomorphism "g", then there exists an "r" in "R" such that "f"("x") = "xr" for all "x" in "M". In the case of non-balanced modules, there will be such an "f" that is not expressible this way.
In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group "M" commuting with all the "R" endomorphisms of "M" are the ones induced by right multiplication by ring elements.
A ring is called balanced if every right "R" module is balanced. It turns out that being balanced is a left-right symmetric condition on rings, and so there is no need to prefix it with "left" or "right".
The study of balanced modules and rings is an outgrowth of the study of QF-1 rings by C.J. Nesbitt and R. M. Thrall. This study was continued in V. P. Camillo's dissertation, and later it became fully developed. The paper gives a particularly broad view with many examples. In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results. A partial list of authors contributing to the theory of balanced modules and rings can be found in the references.