In mathematics, especially
ring theory, the class of Frobenius rings and their generalizations are the extension of work done on
Frobenius algebras. Perhaps the most important generalization is that of quasi-Frobenius rings (QF rings), which are in turn generalized by right pseudo-Frobenius rings (PF rings) and right finitely pseudo-Frobenius rings (FPF rings). Other diverse generalizations of quasi-Frobenius rings include QF-1, QF-2 and QF-3 rings.
These types of rings can be viewed as descendants of algebras examined by
Georg Frobenius. A partial list of pioneers in quasi-Frobenius rings includes
R. Brauer,
K. Morita,
T. Nakayama,
C. J. Nesbitt, and
R. M. Thrall.
Definitions
A ring ''R'' is quasi-Frobenius if and only if ''R'' satisfies any of the following equivalent conditions:
# ''R'' is
Noetherian on one side and
self-injective on one side.
# ''R'' is
Artinian on a side and self-injective on a side.
# All right (or all left) ''R'' modules which are
projective are also
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
.
# All right (or all left) ''R'' modules which are injective are also projective.
A Frobenius ring ''R'' is one satisfying any of the following equivalent conditions. Let ''J''=J(''R'') be the
Jacobson radical of ''R''.
# ''R'' is quasi-Frobenius and the
socle as right ''R'' modules.
#''R'' is quasi-Frobenius and
as left ''R'' modules.
# As right ''R'' modules
, and as left ''R'' modules
.
For a commutative ring ''R'', the following are equivalent:
# ''R'' is Frobenius
# ''R'' is quasi-Frobenius
# ''R'' is a finite direct sum of
local artinian rings which have unique
minimal ideal In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring ''R'' is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonzero left ideal of ''R'' containing no other n ...
s. (Such rings are examples of "zero-dimensional
Gorenstein local rings".)
A ring ''R'' is right pseudo-Frobenius if any of the following equivalent conditions are met:
# Every
faithful right ''R'' module is a
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
for the category of right ''R'' modules.
# ''R'' is right self-injective and is a
cogenerator of Mod-''R''.
# ''R'' is right self-injective and is
finitely cogenerated as a right ''R'' module.
# ''R'' is right self-injective and a right
Kasch ring.
# ''R'' is right self-injective,
semilocal and the socle soc(''R''
''R'') is an
essential submodule In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M ...
of ''R''.
# ''R'' is a cogenerator of Mod-''R'' and is a left Kasch ring.
A ring ''R'' is right finitely pseudo-Frobenius if and only if every
finitely generated faithful right ''R'' module is a generator of Mod-''R''.
Thrall's QF-1,2,3 generalizations
In the seminal article , R. M. Thrall focused on three specific properties of (finite-dimensional) QF algebras and studied them in isolation. With additional assumptions, these definitions can also be used to generalize QF rings. A few other mathematicians pioneering these generalizations included
K. Morita and H. Tachikawa.
Following , let ''R'' be a left or right Artinian ring:
*''R'' is QF-1 if all faithful left modules and faithful right modules are
balanced modules.
*''R'' is QF-2 if each indecomposable projective right module and each indecomposable projective left module has a unique minimal submodule. (I.e. they have simple socles.)
*''R'' is QF-3 if the
injective hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in .
Definition
...
s E(''R''
''R'') and E(
''R''''R'') are both projective modules.
The numbering scheme does not necessarily outline a hierarchy. Under more lax conditions, these three classes of rings may not contain each other. Under the assumption that ''R'' is left or right Artinian however, QF-2 rings are QF-3. There is even an example of a QF-1 and QF-3 ring which is not QF-2.
Examples
*Every Frobenius ''k'' algebra is a Frobenius ring.
*Every
semisimple ring is quasi-Frobenius, since all modules are projective and injective. Even more is true however: semisimple rings are all Frobenius. This is easily verified by the definition, since for semisimple rings
and ''J'' = rad(''R'') = 0.
*The
quotient ring is QF for any positive integer ''n''>1.
*Commutative Artinian
serial rings are all Frobenius, and in fact have the additional property that every quotient ring ''R''/''I'' is also Frobenius. It turns out that among commutative Artinian rings, the serial rings are exactly the rings whose (nonzero) quotients are all Frobenius.
*Many exotic PF and FPF rings can be found as examples in
See also
*
Quasi-Frobenius Lie algebra In mathematics, a quasi-Frobenius Lie algebra
:(\mathfrak, ,\,\,,\,\,\,\beta )
over a field k is a Lie algebra
:(\mathfrak, ,\,\,,\,\,\,)
equipped with a nondegenerate skew-symmetric bilinear form
:\beta : \mathfrak\times\mathfrak\to k, wh ...
Notes
The definitions for QF, PF and FPF are easily seen to be categorical properties, and so they are preserved by
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ...
, however being a Frobenius ring ''is not'' preserved.
For one-sided Noetherian rings the conditions of left or right PF both coincide with QF, but FPF rings are still distinct.
A finite-dimensional algebra ''R'' over a field ''k'' is a Frobenius ''k''-algebra if and only if ''R'' is a Frobenius ring.
QF rings have the property that all of their modules can be embedded in a
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
''R'' module. This can be seen in the following way. A module ''M'' embeds into its
injective hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in .
Definition
...
''E''(''M''), which is now also projective. As a projective module, ''E''(''M'') is a summand of a free module ''F'', and so ''E''(''M'') embeds in ''F'' with the inclusion map. By composing these two maps, ''M'' is embedded in ''F''.
Textbooks
*
*
*
*
References
For QF-1, QF-2, QF-3 rings:
*
*
*{{citation
, last=Thrall , first=R.M.
, title=Some generalization of quasi-Frobenius algebras
, journal=Trans. Amer. Math. Soc.
, volume=64
, year=1948
, pages=173–183
, issn=0002-9947
, doi=10.1090/s0002-9947-1948-0026048-0
, doi-access=free
Module theory
Ring theory