A-IF环的刻画

Characterization of A-IF Ring

引入了A-内射模和A-平坦模的定义,由此构造了A-IF环,利用平坦模和内射模给出了AIF环的8个等价命题,得到了环R分别是IF环、A-正则环和正则环的充要条件,即:R是IF环,当且仅当R是A-IF环且A-平坦模的每个内射子模是平坦模;环R是A-正则环,当且仅当R是A-IF环且A-平坦模的子模是A-平坦模;环R是正则环,当且仅当R是A-IF环且A-平坦模的子模是平坦模。

It is aimportant method that rings are characterized by modules. In this paper we intro- duce the definition of A - injective mould and A - fiat mould, and then we construct A - IF ring. We gave eight equivalent propositions of A - IF ring by injective modules and fiat modules, and obtain the necessary and sufficient conditionof R for A - IF ring, A - regular ring and regu- lar ring. That is that ring R is IF ring if and only if R is A - IF ring and any injective sub - mod- ule of A - fiat module is fiat module ; Ring R is A - regular ring if and only if R is A - IF ring, and any sub - module of A - fiat module is A - fiat module ; Ring R is regular ring if and only if R is A - IF ring, and any sub - module of A - fiat module is fiat module.