# Quasi-Frobenius ring

QF-ring

A (left or right) Artinian ring satisfying the annihilator conditions for each left (or right) ideal (respectively, ) (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 is a quasi-Frobenius ring if and only if the mapping defines a duality between the categories of left and right finitely-generated -modules. A finite-dimensional algebra over a field is a quasi-Frobenius ring if and only if each irreducible right summand of the left -module is isomorphic to some minimal left ideal of . And this is equivalent to the self-duality of the lattices of left and right ideals of .

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 the property of being quasi-Frobenius was originally defined in the following way: If is the complete list of primitive idempotents of (that is, for , and for any primitive idempotent , for some ), is the radical of and is the natural homomorphism, then there is a permutation of the set such that where is the socle of the module . The property of being quasi-Frobenius is equivalent also to each of the following properties: 1) is left Noetherian (cf. Noetherian ring), for every right ideal and for any left ideals and ; 2) satisfies the maximum condition for left (or right) annihilator ideals (in particular, if is left and right Noetherian) and is left and right self-injective (cf. Self-injective ring); 3) is right Artinian and left and right self-injective; 4) every injective (projective) left -module is projective (injective) (cf. Projective module; Injective module); 5) every flat left -module is injective (cf. Flat module); 6) is left and right self-injective and right perfect (cf. Perfect ring); 7) is left and right self-injective and each of its right ideals is an annihilator of some finite set in ; 8) is right perfect and every finitely-generated left -module is contained in a projective module; 9) is coherent (cf. Coherent ring), right perfect, and for all finitely-presented left -modules ; 10) satisfies the maximum condition for left annihilators and for all finitely-presented left -modules ; 11) is left and right Artinian and for every finitely-generated left -module the lengths of the modules and are the same; 12) the ring of endomorphisms of each free left -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 -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 of a ring is transfinitely nilpotent (that is, for some transfinite number , where , and for a limit ordinal number ), then is a quasi-Frobenius ring if and only if is left self-injective and all its one-sided ideals are annihilators. A left module over a quasi-Frobenius ring is faithful if and only if it is a generator of the category of left -modules. The group ring is a quasi-Frobenius ring and if and only if is a finite group and is a quasi-Frobenius ring.

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