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)

References

[a1]	F. Kasch, "Modules and rings" , Acad. Press (1982) (Translated from German)
[a2]	H. Tachikawa, "Quasi-Frobenius rings and generalizations, QF-3 and QF-1 rings" , Lect. notes in math. , 351 , Springer (1973)