# 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.

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 ${\displaystyle \mathrm {soc} (R_{R})\cong R/J}$ as right R modules.
2. R is quasi-Frobenius and ${\displaystyle \mathrm {soc} (_{R}R)\cong R/J}$ as left R modules.
3. As right R modules ${\displaystyle \mathrm {soc} (R_{R})\cong R/J}$, and as left R modules ${\displaystyle \mathrm {soc} (_{R}R)\cong R/J}$.

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 ${\displaystyle \mathrm {soc} (R_{R})=\mathrm {soc} (_{R}R)=R}$ and J = rad(R) = 0.
• The quotient ring ${\displaystyle {\frac {\mathbb {Z} }{n\mathbb {Z} }}}$ 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)

## 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: