In the subfield of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

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 In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...

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

''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 In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring ''S'' of a ring ''R'', denoted C''R''(''S'') in this article. I ...

, 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.The definitions of balanced rings and modules appear in , , , and . 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 In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results.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.

Examples and properties

;Examples * Semisimple rings are balanced. * Every nonzero right ideal over a simple ring is balanced. * Every faithful module over a

quasi-Frobenius ring In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings (QF rings), which are in ...

is balanced. * The

for right Artinian rings states that any simple right ''R'' module is balanced. * The paper contains numerous constructions of nonbalanced modules. * It was established in that

uniserial ring In abstract algebra, a uniserial module ''M'' is a module over a ring ''R'', whose submodules are totally ordered by inclusion. This means simply that for any two submodules ''N''1 and ''N''2 of ''M'', either N_1\subseteq N_2 or N_2\subseteq N_1. ...

s are balanced. Conversely, a balanced ring which is finitely generated as a module over its center is uniserial. * Among commutative Artinian rings, the balanced rings are exactly the

s. ;Properties * Being "balanced" is a categorical property for modules, that is, it is preserved by

Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modul ...

. Explicitly, if ''F''(–) is a Morita equivalence from the category of ''R'' modules to the category of ''S'' modules, and if ''M'' is balanced, then ''F''(''M'') is balanced. * The structure of balanced rings is also completely determined in , and is outlined in . * In view of the last point, the property of being a balanced ring is a Morita invariant property. * The question of which rings have all finitely generated right ''R'' modules balanced has already been answered. This condition turns out to be equivalent to the ring ''R'' being balanced.

Notes

References

* * * * * * * *{{citation , last1=Nesbitt, first1=C. J. , last2=Thrall , first2=R. M. , title=Some ring theorems with applications to modular representations , journal=Ann. of Math. , series= 2 , volume=47 , issue=3 , year=1946 , pages=551–567 , issn=0003-486X , mr=0016760 , doi=10.2307/1969092, jstor=1969092 Module theory Ring theory