摘要
本文研究了较ＰＦ—环更广泛的一类环，被称为ｆＰＦ—环，得到如下结果：（１）Ｒ是ｆＰＦ—环当且仅当对于每个ａ∈Ｒ，存在ｆ∈Ｈ（Ｒ），使得ａｎｎ＿Ｒ（ｆ（ａ））是Ｒ的一个纯理想；（２）设Ｒ是局部环，则Ｒ是ｆＰＦ—环当且仅当对于每个ａ∈Ｒ，存在ｆ∈Ｈ（Ｒ）使得ｆ（ａ）＝０或者ｆ（ａ）不是零因子；（３）Ｒ是ｆＰＦ—环当且仅当对每个ａ∈Ｒ。存在ｆ∈Ｈ（Ｒ）使得ｆ（ａ）在每个局部化Ｒｐ中不是零因子，或者在每个Ｒｐ中ｆ（ａ）＝０；（４）设Ｒ是强ｆＰＦ—环，且对于ｘ∈Ｒ，ａ∈ａｎｎ＿Ｒ（ｘ）当且仅当ｆ（ａ）∈ａｎｎ＿Ｒ（ｘ）对某ｆ∈Ｈ（Ｒ），则Ｒ是ＰＦ—环，另外。
In this paper, we investigate more general rings than PF rings, called fPF rings.We prove the following results:(1) R is a fPF ring if and only if for each a∈R there exists f∈H(R) such that ann_R(f(a)) is a pure ideal of R. (2) Let R be a local ring. Then R is a fPF ring if and only if for every a∈R, there exists f∈H(R) such that f(a) is a non zero divisor or f(a)=0. (3) Ring R is a fPF ring if and only if for every a∈R, there is f∈H(R) such that f(a) is a non zero divisor in each localization Rp or f(a) =0 in each Rp. (4) Ring is a strongly fPF ring, for x∈R, a∈ann_R (x) if and only if f(a)∈ann_R(x), then R is a PF ring. We also provide an example of fPF ring which is not a PF ring.
基金
安徽省教委科研资金资助
关键词
fPF环
交换环
广义
PF环
FPF ring, pure ideal, PF ring, strongly fPF ring.
