View source for Quasi-Frobenius ring

''QF-ring''

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

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for each left (or right) ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765002.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765003.png" />) (see [[Annihilator|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765004.png" /> is a quasi-Frobenius ring if and only if the mapping

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defines a duality between the categories of left and right finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765006.png" />-modules. A finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765007.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765008.png" /> is a quasi-Frobenius ring if and only if each irreducible right summand of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765009.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650010.png" /> is isomorphic to some minimal left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650011.png" />. And this is equivalent to the self-duality of the lattices of left and right ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650012.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650013.png" /> the property of being quasi-Frobenius was originally defined in the following way: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650014.png" /> is the complete list of primitive idempotents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650015.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650017.png" />, and for any primitive idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650019.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650020.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650021.png" /> is the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650023.png" /> is the natural homomorphism, then there is a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650024.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650025.png" /> such that

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650027.png" /> is the [[Socle|socle]] of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650028.png" />. The property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650029.png" /> being quasi-Frobenius is equivalent also to each of the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650030.png" /> is left Noetherian (cf. [[Noetherian ring|Noetherian ring]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650031.png" /> for every right ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650032.png" /> and

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for any left ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650035.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650036.png" /> satisfies the maximum condition for left (or right) annihilator ideals (in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650037.png" /> is left and right Noetherian) and is left and right self-injective (cf. [[Self-injective ring|Self-injective ring]]); 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650038.png" /> is right Artinian and left and right self-injective; 4) every injective (projective) left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650039.png" />-module is projective (injective) (cf. [[Projective module|Projective module]]; [[Injective module|Injective module]]); 5) every flat left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650040.png" />-module is injective (cf. [[Flat module|Flat module]]); 6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650041.png" /> is left and right self-injective and right perfect (cf. [[Perfect ring|Perfect ring]]); 7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650042.png" /> is left and right self-injective and each of its right ideals is an annihilator of some finite set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650043.png" />; 8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650044.png" /> is right perfect and every finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650045.png" />-module is contained in a projective module; 9) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650046.png" /> is coherent (cf. [[Coherent ring|Coherent ring]]), right perfect, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650047.png" /> for all finitely-presented left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650048.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650049.png" />; 10) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650050.png" /> satisfies the maximum condition for left annihilators and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650051.png" /> for all finitely-presented left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650052.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650053.png" />; 11) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650054.png" /> is left and right Artinian and for every finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650055.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650056.png" /> the lengths of the modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650058.png" /> are the same; 12) the ring of endomorphisms of each free left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650059.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650060.png" />-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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,   "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Kasch,   "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Tachikawa,   "Quasi-Frobenius rings and generalizations, QF-3 and QF-1 rings" , ''Lect. notes in math.'' , '''351''' , Springer  (1973)</TD></TR></table>