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''QF-ring''
 
''QF-ring''
  
 
A (left or right) [[Artinian ring|Artinian ring]] satisfying the annihilator conditions
 
A (left or right) [[Artinian ring|Artinian ring]] satisfying the annihilator conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765001.png" /></td> </tr></table>
+
$$
 +
\mathfrak Z _ {l} ( \mathfrak Z _ {r} ( L) )  = L \ \
 +
\textrm{ and } \  \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) )  = H
 +
$$
  
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
+
for each left (or right) ideal $  L $(
 +
respectively, $  H $)  
 +
(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 $  R $
 +
is a quasi-Frobenius ring if and only if the mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q0765005.png" /></td> </tr></table>
+
$$
 +
M  \mapsto  \mathop{\rm Hom} _ {R} ( M , R )
 +
$$
  
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" />.
+
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 <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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650026.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Soc} ( e _ {i} R )  \cong  \phi ( e _ {\pi ( i) }  R ) \ \
 +
\textrm{ and } \  \mathop{\rm Soc} ( R e _ {\pi ( i) }  )  \cong  \phi ( R e _ {i} ) ,
 +
$$
  
where <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
+
where $  \mathop{\rm Soc}  M $
 +
is the [[Socle|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|Noetherian ring]]), $  \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H $
 +
for every right ideal $  H $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076500/q07650033.png" /></td> </tr></table>
+
$$
 +
\mathfrak Z _ {r} ( L _ {1} \cap L _ {2} )  = \
 +
\mathfrak Z _ {r} ( L _ {1} ) + \mathfrak Z _ {r} ( L _ {2} )
 +
$$
  
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.
+
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|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|Projective module]]; [[Injective module|Injective module]]); 5) every flat left $  R $-
 +
module is injective (cf. [[Flat module|Flat module]]); 6) $  R $
 +
is left and right self-injective and right perfect (cf. [[Perfect ring|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|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|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]] $  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-<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- $  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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:09, 6 June 2020


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)

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