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 term ''a ...

known as

module theory In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...

, 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 gr ...

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 ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...

s are balanced. * Every nonzero right

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

over a

simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a sim ...

is balanced. * Every

faithful module In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of . Over an integral domain, a module that has a nonzero annihilator is a ...

over a quasi-Frobenius ring is balanced. * The

for right Artinian rings states that any

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

right ''R'' module is balanced. * The paper contains numerous constructions of nonbalanced modules. * It was established in that uniserial rings are balanced. Conversely, a balanced ring which is finitely generated as a module over its

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

is uniserial. * Among commutative Artinian rings, the balanced rings are exactly the quasi-Frobenius rings. ;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 modules ...

. 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