We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.
Let R be a commutative ring with identity. An R-module M is said to be a comultiplication module if for every submodule N of M, there exists an ideal I of R such that N = （0：M I）. In this paper, we show： （1） If ...Let R be a commutative ring with identity. An R-module M is said to be a comultiplication module if for every submodule N of M, there exists an ideal I of R such that N = （0：M I）. In this paper, we show： （1） If M is a comultiplication module and N is a copure submodule of M, then M/N is a comultiplication module. （2） If M is a comultiplication module satisfying the DAC and N ≤ M, then N ≤eM if and only if there exists I ≤ R such that N = （0 ：M I）. （3） If M is a comultiplication module satisfying the DAC, then M is finitely cogenerated. Finally, we give a partial answer to a question posed by Ansari-Toroghy and Farshadifar.展开更多
R是任意一个结合环,M既是左R-模又是右R-模。M称为强余纯内射的,如果对于任意的内射R-模E和任意的i≥1都有Ext^i(E,M)=0;如果Ext^1(E,M)=0,我们称M是余纯内射的。类似的,M称为余纯平坦的,如果对于任意的内射R-模E和任意的i≥1都有Tor_i(...R是任意一个结合环,M既是左R-模又是右R-模。M称为强余纯内射的,如果对于任意的内射R-模E和任意的i≥1都有Ext^i(E,M)=0;如果Ext^1(E,M)=0,我们称M是余纯内射的。类似的,M称为余纯平坦的,如果对于任意的内射R-模E和任意的i≥1都有Tor_i(E,M)=0;如果Tor_1(E,M)=0,我们称M是余纯平坦的。我们找出并证明了(强)余纯内射模和(强)余纯平坦模之间的关系。更重要的是,我们给出了由Enochs and Jenda所列出的一些重要结论的证明。展开更多
文摘We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.
文摘Let R be a commutative ring with identity. An R-module M is said to be a comultiplication module if for every submodule N of M, there exists an ideal I of R such that N = （0：M I）. In this paper, we show： （1） If M is a comultiplication module and N is a copure submodule of M, then M/N is a comultiplication module. （2） If M is a comultiplication module satisfying the DAC and N ≤ M, then N ≤eM if and only if there exists I ≤ R such that N = （0 ：M I）. （3） If M is a comultiplication module satisfying the DAC, then M is finitely cogenerated. Finally, we give a partial answer to a question posed by Ansari-Toroghy and Farshadifar.
文摘R是任意一个结合环,M既是左R-模又是右R-模。M称为强余纯内射的,如果对于任意的内射R-模E和任意的i≥1都有Ext^i(E,M)=0;如果Ext^1(E,M)=0,我们称M是余纯内射的。类似的,M称为余纯平坦的,如果对于任意的内射R-模E和任意的i≥1都有Tor_i(E,M)=0;如果Tor_1(E,M)=0,我们称M是余纯平坦的。我们找出并证明了(强)余纯内射模和(强)余纯平坦模之间的关系。更重要的是,我们给出了由Enochs and Jenda所列出的一些重要结论的证明。
