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.
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.
soc(RR)\congR/J
soc(RR)\congR/J
soc(RR)\congR/J
soc(RR)\congR/J
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.
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.
soc(RR)=soc(RR)=R
Z/nZ
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.
For QF-1, QF-2, QF-3 rings: