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, which are in turn generalized by right pseudo-Frobenius rings and right finitely pseudo-Frobenius rings. Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings.
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

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 R modules which are projective are also injective.
  4. All right R modules which are injective are also projective.
A Frobenius ring R is one satisfying any of the following equivalent conditions. Let J=J be the Jacobson radical of R.
  1. R is quasi-Frobenius and the socle as right R modules.
  2. R is quasi-Frobenius and as left R modules.
  3. As right R modules, and as left R modules.
For a commutative ring R, the following are equivalent:
  1. R is Frobenius
  2. R is quasi-Frobenius
  3. R is a finite direct sum of local artinian rings which have unique minimal ideals.
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 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, R. M. Thrall focused on three specific properties of 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, let R be a left or right Artinian ring:
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