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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quasi-Frobenius ring ： ウィキペディア英語版 Quasi-Frobenius ring 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== 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. 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. # 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 $\backslash mathrm(R\_R)\backslash cong\; R/J$ as right ''R'' modules. #''R'' is quasi-Frobenius and $\backslash mathrm(\_R\; R)\backslash cong\; R/J$ as left ''R'' modules. # As right ''R'' modules $\backslash mathrm(R\_R)\backslash cong\; R/J$, and as left ''R'' modules $\backslash mathrm(\_R\; R)\backslash cong\; R/J$. For a commutative ring ''R'', the following are equivalent: # ''R'' is Frobenius # ''R'' is QF # ''R'' is a finite direct sum of local artinian rings which have unique minimal ideals. (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 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 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quasi-Frobenius ring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.