Girish Mahajan (Editor)

Quasi Frobenius ring

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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 turn generalized by right pseudo-Frobenius rings (PF rings) and right finitely pseudo-Frobenius rings (FPF rings). Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings.

Contents

These types of rings can be viewed as descendants of algebras examined by Georg Frobenius. A partial list of pioneers in quasi-Frobenius rings includes R. Brauer, K. Morita, T. Nakayama, C. J. Nesbitt, and R. M. Thrall.

Definitions

For the sake of presentation, it will be easier to define quasi-Frobenius rings first. In the following characterizations of each type of ring, many properties of the ring will be revealed.

A ring R is quasi-Frobenius if and only if R satisfies any of the following equivalent conditions:

  1. R is Noetherian on one side and self-injective on one side.
  2. R is Artinian on a side and self-injective on a side.
  3. All right (or all left) R modules which are projective are also injective.
  4. All right (or all left) R modules which are injective are also projective.

A Frobenius ring R is one satisfying any of the following equivalent conditions. Let J=J(R) be the Jacobson radical of R.

  1. R is quasi-Frobenius and the socle s o c ( R R ) ≅ R / J as right R modules.
  2. R is quasi-Frobenius and s o c ( R R ) ≅ R / J as left R modules.
  3. As right R modules s o c ( R R ) ≅ R / J , and as left R modules s o c ( R R ) ≅ R / J .

For a commutative ring R, the following are equivalent:

  1. R is Frobenius
  2. R is QF
  3. R is a finite direct sum of local artinian rings which have unique minimal ideals. (Such rings are examples of "zero-dimensional Gorenstein local rings".)

A ring R is right pseudo-Frobenius if any of the following equivalent conditions are met:

  1. Every faithful right R module is a generator for the category of right R modules.
  2. R is right self-injective and is a cogenerator of Mod-R.
  3. R is right self-injective and is finitely cogenerated as a right R module.
  4. R is right self-injective and a right Kasch ring.
  5. R is right self-injective, semilocal and the socle soc(RR) is an essential submodule of R.
  6. R is a cogenerator of Mod-R and is a left Kasch ring.

A ring R is right finitely pseudo-Frobenius if and only if every finitely generated faithful right R module is a generator of Mod-R.

Thrall's QF-1,2,3 generalizations

In the seminal article (Thrall 1948), R. M. Thrall focused on three specific properties of (finite-dimensional) QF algebras and studied them in isolation. With additional assumptions, these definitions can also be used to generalize QF rings. A few other mathematicians pioneering these generalizations included K. Morita and H. Tachikawa.

Following (Anderson & Fuller 1992), let R be a left or right Artinian ring:

  • R is QF-1 if all faithful left modules and faithful right modules are balanced modules.
  • R is QF-2 if each indecomposable projective right module and each indecomposable projective left module has a unique minimal submodule. (I.e. they have simple socles.)
  • R is QF-3 if the injective hulls E(RR) and E(RR) are both projective modules.

    • The numbering scheme does not necessarily outline a hierarchy. Under more lax conditions, these three classes of rings may not contain each other. Under the assumption that R is left or right Artinian however, QF-2 rings are QF-3. There is even an example of a QF-1 and QF-3 ring which is not QF-2.

    Examples

  • Every Frobenius k algebra is a Frobenius ring.
  • Every semisimple ring is clearly quasi-Frobenius, since all modules are projective and injective. Even more is true however: semisimple rings are all Frobenius. This is easily verified by the definition, since for semisimple rings s o c ( R R ) = s o c ( R R ) = R and J = rad(R) = 0.
  • The quotient ring Z n Z is QF for any positive integer n>1.
  • Commutative Artinian serial rings are all Frobenius, and in fact have the additional property that every quotient ring R/I is also Frobenius. It turns out that among commutative Artinian rings, the serial rings are exactly the rings whose (nonzero) quotients are all Frobenius.
  • Many exotic PF and FPF rings can be found as examples in (Faith 1984)

    • Textbooks

  • Anderson, Frank Wylie; Fuller, Kent R (1992), Rings and Categories of Modules, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97845-1 
  • Faith, Carl; Page, Stanley (1984), FPF Ring Theory: Faithful modules and generators of Mod-$R$, London Mathematical Society Lecture Note Series No. 88, Cambridge University Press, ISBN 0-521-27738-8, MR 0754181 
  • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 
  • Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge University Press, ISBN 0-521-81593-2 

    • References

    Quasi-Frobenius ring Wikipedia
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