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 (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.
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:
A Frobenius ring "R" is one satisfying any of the following equivalent conditions. Let "J"=J("R") be the Jacobson radical of "R".
For a commutative ring "R", the following are equivalent:
A ring "R" is right pseudo-Frobenius if any of the following equivalent conditions are met:
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 (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 , 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.
Notes.
The definitions for QF, PF and FPF are easily seen to be categorical properties, and so they are preserved by Morita equivalence, however being a Frobenius ring "is not" preserved.
For one-sided Noetherian rings the conditions of left or right PF both coincide with QF, but FPF rings are still distinct.
A finite-dimensional algebra "R" over a field "k" is a Frobenius "k"-algebra if and only if "R" is a Frobenius ring.
QF rings have the property that all of their modules can be embedded in a free "R" module. This can be seen in the following way. A module "M" embeds into its injective hull "E"("M"), which is now also projective. As a projective module, "E"("M") is a summand of a free module "F", and so "E"("M") embeds in "F" with the inclusion map. By composing these two maps, "M" is embedded in "F".
References.
For QF-1, QF-2, QF-3 rings: