Balanced module

In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module if every R-homomorphism from M into M is given by multiplication by a ring element. Explicitly, for any R-endomorphism f, there exists an r in R such that f(x)=xr for all x in M.

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 (Dlab & Ringel 1972) 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 In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results.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 mode

• The Double centralizer theorem for right Artinian rings states that any simple right R module is balanced.
• The paper (Dlab & Ringel 1972) contains numerous constructions of nonbalanced modules.
• It was established in (Nesbitt & Thrall 1946) that uniserial rings 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 quasi-Frobenius rings.

Properties

• Being "balanced" is a categorical property for modules, that is, it is preserved by Morita equivalence. 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 (Dlab & Ringel 1972), and is outlined in (Faith 1999, p.222-224).
• 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.

References

• Camillo, Victor P. (1970), "Balanced rings and a problem of Thrall", Trans. Amer. Math. Soc., 149: 143–153, ISSN 0002-9947, MR 0260794

• Cunningham, R. S.; Rutter, E. A., Jr. (1972), "The double centralizer property is categorical", Rocky Mountain J. Math., 2 (4): 627–629, ISSN 0035-7596, MR 0310017{{citation}}: CS1 maint: multiple names: authors list (link)

• Dlab, Vlastimil; Ringel, Claus Michael (1972), "Rings with the double centralizer property", J. Algebra, 22: 480–501, ISSN 0021-8693, MR 0306258

• Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, ISBN 0-8218-0993-8, MR 1657671

• Nesbitt, C. J.; Thrall, R. M. (1946), "Some ring theorems with applications to modular representations", Ann. of Math. (2), 47: 551–567, ISSN 0003-486X, MR 0016760