# Quasi-Frobenius ring

QF-ring

A (left or right) Artinian ring satisfying the annihilator conditions

$$\mathfrak Z _ {l} ( \mathfrak Z _ {r} ( L) ) = L \ \ \textrm{ and } \ \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H$$

for each left (or right) ideal $L$( respectively, $H$) (see Annihilator). A left Artinian ring that satisfies only one of these annihilator conditions need not be a quasi-Frobenius ring. Quasi-Frobenius rings are of interest because of the presence of duality: A left Artinian ring $R$ is a quasi-Frobenius ring if and only if the mapping

$$M \mapsto \mathop{\rm Hom} _ {R} ( M , R )$$

defines a duality between the categories of left and right finitely-generated $R$- modules. A finite-dimensional algebra $A$ over a field $P$ is a quasi-Frobenius ring if and only if each irreducible right summand of the left $A$- module $\mathop{\rm Hom} _ {P} ( A _ {A} , P )$ is isomorphic to some minimal left ideal of $A$. And this is equivalent to the self-duality of the lattices of left and right ideals of $A$.

Quasi-Frobenius rings were introduced as a generalization of Frobenius algebras, determined by the requirement that the right and left regular representations are equivalent. For a left and right Artinian ring $R$ the property of being quasi-Frobenius was originally defined in the following way: If $e _ {1} \dots e _ {n}$ is the complete list of primitive idempotents of $R$( that is, $R e _ {i} \Nsm R e _ {j}$ for $i \neq j$, and for any primitive idempotent $e$, $R e \cong R e _ {i}$ for some $i$), $J$ is the radical of $R$ and $\phi : R \rightarrow R / J$ is the natural homomorphism, then there is a permutation $\pi$ of the set $\{ 1 \dots n \}$ such that

$$\mathop{\rm Soc} ( e _ {i} R ) \cong \phi ( e _ {\pi ( i) } R ) \ \ \textrm{ and } \ \mathop{\rm Soc} ( R e _ {\pi ( i) } ) \cong \phi ( R e _ {i} ) ,$$

where $\mathop{\rm Soc} M$ is the socle of the module $M$. The property of $R$ being quasi-Frobenius is equivalent also to each of the following properties: 1) $R$ is left Noetherian (cf. Noetherian ring), $\mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H$ for every right ideal $H$ and

$$\mathfrak Z _ {r} ( L _ {1} \cap L _ {2} ) = \ \mathfrak Z _ {r} ( L _ {1} ) + \mathfrak Z _ {r} ( L _ {2} )$$

for any left ideals $L _ {1}$ and $L _ {2}$; 2) $R$ satisfies the maximum condition for left (or right) annihilator ideals (in particular, if $R$ is left and right Noetherian) and is left and right self-injective (cf. Self-injective ring); 3) $R$ is right Artinian and left and right self-injective; 4) every injective (projective) left $R$- module is projective (injective) (cf. Projective module; Injective module); 5) every flat left $R$- module is injective (cf. Flat module); 6) $R$ is left and right self-injective and right perfect (cf. Perfect ring); 7) $R$ is left and right self-injective and each of its right ideals is an annihilator of some finite set in $R$; 8) $R$ is right perfect and every finitely-generated left $R$- module is contained in a projective module; 9) $R$ is coherent (cf. Coherent ring), right perfect, and $\mathop{\rm Ext} _ {R} ( M , R ) = 0$ for all finitely-presented left $R$- modules $M$; 10) $R$ satisfies the maximum condition for left annihilators and $\mathop{\rm Ext} _ {R} ( M , R ) = 0$ for all finitely-presented left $R$- modules $M$; 11) $R$ is left and right Artinian and for every finitely-generated left $R$- module $M$ the lengths of the modules $M$ and $\mathop{\rm Hom} _ {R} ( M , R )$ are the same; 12) the ring of endomorphisms of each free left $R$- module is left self-injective; or 13) finitely-generated one-sided ideals of the ring of endomorphisms of a projective generator (injective co-generator) of the category of left $R$- modules are annihilators.

Injective modules over a quasi-Frobenius ring split into a direct sum of cyclic modules. For commutative rings the converse is also true. If the Jacobson radical $J$ of a ring $R$ is transfinitely nilpotent (that is, $J ^ \alpha = 0$ for some transfinite number $\alpha$, where $J ^ {1} = J$, $J ^ \alpha = J ^ {\alpha - 1 } J$ and $J ^ \alpha = \cap _ {\beta < \alpha } J ^ \beta$ for a limit ordinal number $\alpha$), then $R$ is a quasi-Frobenius ring if and only if $R$ is left self-injective and all its one-sided ideals are annihilators. A left module over a quasi-Frobenius ring $R$ is faithful if and only if it is a generator of the category of left $R$- modules. The group ring $R G$ is a quasi-Frobenius ring and if and only if $G$ is a finite group and $R$ is a quasi-Frobenius ring.

Certain generalizations of quasi-Frobenius rings have also been studied; a left QF- $3$- ring $R$ is defined by the requirement that there exists a faithful left $R$- module that is contained as a direct summand in any faithful left $R$- module; a left QF- $3 ^ \prime$- ring $R$ is defined by the requirement that the injective hull of the left $R$- module $R$ can be imbedded in the direct product of some set of copies of $R$. A left pseudo-Frobenius ring (or left PF-ring) is defined by each of the following properties: a) $R$ is an injective co-generator of the category of left $R$- modules; b) every faithful left $R$- module is a generator of the category of left $R$- modules; or c) $R$ is a left QF- $3$- ring and the annihilator of any right ideal different from $R$ is non-zero.

#### References

 [1] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) [2] T.S. Tol'skaya, "Quasi-Frobenius rings and their generalizations" L.A. Skornyakov (ed.) A.V. Mikhalev (ed.) , Modules , 2 , Novosibirsk (1973) pp. 42–48 (In Russian) [3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)