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 right balanced if every right R module is balanced.

The study of balanced modules and rings is an outgrowth of R. M. Thrall's study of QF-1 rings. An incomplete list of authors contributing to the theory of balanced modules includes (Camillo 1970), (Cunningham & Rutter 1972), (Dlab & Ringel 1972) and (Faith 1999).

Examples and properties

Examples

• (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.

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 Morita invariant.
• 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