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| In mathematics, especially [[ring theory]], the class of '''Frobenius rings''' and their generalizations are the extension of work done on [[Frobenius algebra]]s. 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.
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| 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 [[Richard Brauer|R. Brauer]], [[Kiiti Morita|K. Morita]], [[Tadashi Nakayama (mathematician)|T. Nakayama]], [[Cecil J. Nesbitt|C. J. Nesbitt]], and [[Robert M. Thrall|R. M. Thrall]].
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| ==Definitions==
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| For the sake of presentation, it will be easier to define quasi-Frobenius rings first. In the following characterizations of each type of ring, many properties of the ring will be revealed.
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| A ring ''R'' is '''quasi-Frobenius''' if and only if ''R'' satisfies any of the following equivalent conditions:
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| # ''R'' is [[noetherian ring|Noetherian]] on one side and [[injective module#Self-injective rings|self-injective]] on one side.
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| # ''R'' is [[Artinian ring|Artinian]] on a side and self-injective on a side.
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| # All right (or all left) ''R'' modules which are [[projective module|projective]] are also [[injective module|injective]].
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| # All right (or all left) ''R'' modules which are injective are also projective.
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| A '''Frobenius ring''' ''R'' is one satisfying any of the following equivalent conditions. Let ''J''=J(''R'') be the [[Jacobson radical]] of ''R''.
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| # ''R'' is quasi-Frobenius and the [[Socle (mathematics)|socle]] <math>\mathrm{soc}(R_R)\cong R/J</math> as right ''R'' modules.
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| #''R'' is quasi-Frobenius and <math>\mathrm{soc}(_R R)\cong R/J</math> as left ''R'' modules.
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| # As right ''R'' modules <math>\mathrm{soc}(R_R)\cong R/J</math>, and as left ''R'' modules <math>\mathrm{soc}(_R R)\cong R/J</math>.
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| For a commutative ring ''R'', the following are equivalent:
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| # ''R'' is Frobenius
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| # ''R'' is QF
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| # ''R'' is a finite direct sum of [[local ring|local]] artinian rings which have unique [[minimal ideal]]s. (Such rings are examples of "zero-dimensional [[Gorenstein ring|Gorenstein local rings]]".)
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| A ring ''R'' is '''right pseudo-Frobenius''' if any of the following equivalent conditions are met:
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| # Every [[faithful module|faithful]] right ''R'' module is a [[generator (category theory)|generator]] for the category of right ''R'' modules.
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| # ''R'' is right self-injective and is a [[Injective cogenerator|cogenerator]] of Mod-''R''.
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| # ''R'' is right self-injective and is [[finitely generated module|finitely cogenerated]] as a right ''R'' module.
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| # ''R'' is right self-injective and a right [[Kasch ring]].
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| # ''R'' is right self-injective, [[semilocal ring|semilocal]] and the socle soc(''R''<sub>''R''</sub>) is an [[essential submodule]] of ''R''.
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| # ''R'' is a cogenerator of Mod-''R'' and is a left Kasch ring.
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| A ring ''R'' is '''right finitely pseudo-Frobenius''' if and only if every [[Finitely generated module|finitely generated]] faithful right ''R'' module is a generator of Mod-''R''.
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| ==Thrall's QF-1,2,3 generalizations==
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| In the seminal article {{harv|Thrall|1948}}, 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 [[Kiiti Morita|K. Morita]] and H. Tachikawa.
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| Following {{harv|Anderson|Fuller|1992}}, let ''R'' be a left or right Artinian ring:
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| *''R'' is QF-1 if all faithful left modules and faithful right modules are [[balanced module]]s.
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| *''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.)
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| *''R'' is QF-3 if the [[injective hull]]s E(''R''<sub>''R''</sub>) and E(<sub>''R''</sub>''R'') are both projective modules.
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| 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.
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| ==Examples==
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| *Every Frobenius ''k'' algebra is a Frobenius ring.
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| *Every [[semisimple ring]] is clearly 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 <math>\mathrm{soc}(R_R)=\mathrm{soc}(_R R)=R</math> and ''J'' = rad(''R'') = 0.
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| *The [[quotient ring]] <math>\frac{ \mathbb{Z} }{ n \mathbb{Z} }</math> is QF for any positive integer ''n''>1.
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| *Commutative Artinian [[serial module|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.
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| *Many exotic PF and FPF rings can be found as examples in {{harv|Faith|1984}}
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| ==See also==
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| *[[Quasi-Frobenius Lie algebra]]
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| ==Notes==
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| The definitions for QF, PF and FPF are easily seen to be categorical properties, and so they are preserved by [[Morita equivalence]], however being a Frobenius ring ''is not'' preserved.
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| For one-sided Noetherian rings the conditions of left or right PF both coincide with QF, but FPF rings are still distinct.
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| A finite dimensional algebra ''R'' over a field ''k'' is a Frobenius ''k''-algebra if and only if ''R'' is a Frobenius ring.
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| QF rings have the property that all of their modules can be embedded in a [[free module|free]] ''R'' module. This can be seen in the following way. A module ''M'' embeds into its [[injective hull]] ''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''.
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| ==Textbooks==
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| *{{Citation | last1=Anderson | first1=Frank Wylie | last2=Fuller | first2=Kent R | title=Rings and Categories of Modules | url=http://books.google.com/?id=PswhrD_wUIkC | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-97845-1 | year=1992}}
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| *{{Citation | last1=Faith| first1=Carl| last2=Page| first2=Stanley | title=FPF Ring Theory: Faithful modules and generators of Mod-$R$| publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series No. 88 | isbn=0-521-27738-8 | id={{MathSciNet|id=0754181}} | year=1984}}
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | id={{MathSciNet|id=1653294}} | year=1999}}
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| *{{Citation | last1=Nicholson | first1=W. K. | last2=Yousif | first2=M. F. | title=Quasi-Frobenius rings | publisher=Cambridge University Press | isbn=0-521-81593-2
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| | year=2003}}
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| ==References==
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| For QF-1, QF-2, QF-3 rings:
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| *{{citation
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| |author=Morita, Kiiti
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| |title=On algebras for which every faithful representation is its own second commutator
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| |journal=Math. Z.
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| |volume=69
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| |year=1958
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| |pages=429–434
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| |issn=0025-5874
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| }}
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| *{{citation
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| |author=Ringel, Claus Michael
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| |author=Tachikawa, Hiroyuki
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| |title=${\rm QF}-3$ rings
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| |journal=J. Reine Angew. Math.
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| |volume=272
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| |year=1974
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| |pages=49–72
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| |issn=0075-4102
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| }}
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| *{{citation
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| |author=Thrall, R. M.
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| |title=Some generalization of quasi-Frobenius algebras
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| |journal=Trans. Amer. Math. Soc.
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| |volume=64
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| |year=1948
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| |pages=173–183
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| |issn=0002-9947
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| }}
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| [[Category:Module theory]]
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| [[Category:Ring theory]]
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