Difference between revisions of "Quasi-Frobenius ring"
(Importing text file) |
m (link) |
||
Line 23: | Line 23: | ||
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. |
====References==== | ====References==== |
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) |
Quasi-Frobenius ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-Frobenius_ring&oldid=39563