View source for Quasi-Frobenius ring

<!--
q0765001.png
$#A+1 = 91 n = 0
$#C+1 = 91 : ~/encyclopedia/old_files/data/Q076/Q.0706500 Quasi\AAhFrobenius ring,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''QF-ring''

A (left or right) [[Artinian ring|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|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|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

$$ 
\mathfrak Z _ {r} ( L _ {1} \cap L _ {2} )  = \ 
\mathfrak Z _ {r} ( L _ {1} ) + \mathfrak Z _ {r} ( L _ {2} )
$$

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|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]]  $  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====
<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>