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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|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650061.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650062.png" /> is transfinitely nilpotent (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650063.png" /> for some transfinite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650067.png" /> for a limit ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650068.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650069.png" /> is a quasi-Frobenius ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650070.png" /> is left self-injective and all its one-sided ideals are annihilators. A left module over a quasi-Frobenius ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650071.png" /> is faithful if and only if it is a generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650072.png" />-modules. The group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650073.png" /> is a quasi-Frobenius ring and if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650074.png" /> is a finite group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650075.png" /> is a quasi-Frobenius ring.
 
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|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650061.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650062.png" /> is transfinitely nilpotent (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650063.png" /> for some transfinite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650067.png" /> for a limit ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650068.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650069.png" /> is a quasi-Frobenius ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650070.png" /> is left self-injective and all its one-sided ideals are annihilators. A left module over a quasi-Frobenius ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650071.png" /> is faithful if and only if it is a generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650072.png" />-modules. The group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650073.png" /> is a quasi-Frobenius ring and if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650074.png" /> is a finite group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650075.png" /> is a quasi-Frobenius ring.
  
Certain generalizations of quasi-Frobenius rings have also been studied; a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650077.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650078.png" /> is defined by the requirement that there exists a faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650079.png" />-module that is contained as a direct summand in any faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650080.png" />-module; a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650082.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650083.png" /> is defined by the requirement that the injective hull of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650084.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650085.png" /> can be imbedded in the direct product of some set of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650086.png" />. A left pseudo-Frobenius ring (or left PF-ring) is defined by each of the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650087.png" /> is an injective co-generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650088.png" />-modules; b) every faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650089.png" />-module is a generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650090.png" />-modules; or c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650091.png" /> is a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650092.png" />-ring and the annihilator of any right ideal different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650093.png" /> is non-zero.
+
Certain generalizations of quasi-Frobenius rings have also been studied; a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650077.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650078.png" /> is defined by the requirement that there exists a faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650079.png" />-module that is contained as a direct summand in any faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650080.png" />-module; a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650082.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650083.png" /> is defined by the requirement that the [[injective hull]] of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650084.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650085.png" /> can be imbedded in the direct product of some set of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650086.png" />. A left pseudo-Frobenius ring (or left PF-ring) is defined by each of the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650087.png" /> is an injective co-generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650088.png" />-modules; b) every faithful left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650089.png" />-module is a generator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650090.png" />-modules; or c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650091.png" /> is a left QF-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650092.png" />-ring and the annihilator of any right ideal different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650093.png" /> is non-zero.
  
 
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Revision as of 20:29, 30 October 2016

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.

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)


Comments

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)
How to Cite This Entry:
Quasi-Frobenius ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-Frobenius_ring&oldid=39563
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article