拓扑空间上某些函数环的共同性质
Common properties of some function rings on a topological space
摘要：
对非空拓扑空间X，用$F(X)$表示$X$上所有具有点式加法和乘法的实值函数组成的环，用$C(X)$表示$F(X)$的连续成员.设$A(X)$是$F(X)$的一个子环，$B$是$A(X)$的一个非零非空子集.证明了存在$X$的子集$S$和环同态$ϕ:A(X)\to A(S)$使得$ker ϕ=Ann(B)$ . $F(X)$的格序子环$A(X)$称为$P$ -凸的，如果$A(X)$的每个素理想都是$A(X)$中的绝对凸理想.研究了$F(X)$的$P$ -凸子环的一些性质.证明了$X$上的Baire 1函数环是$P$ -凸的. $A(X)$中的真理想$I$称为伪固定理想，如果$\bigcap \overline{Z[I]}\neq \emptyset $，其中$ \overline{Z[I]}=\{cl_X f^{-1}(0) | f\in I\}$ .给出了$F(X)$的某些子环中伪不动理想的一些刻画。设$X$是完全正则Hausdorff空间，$A(X)$是$F(X)$的子环，使得$f \in A(X)$是$A(X) $的单位当且仅当$f^{-1}(0)=\emptyset$和$C(X) \subseteq F(X)$ .证明了$A(X)$是Gelfand环，$X$是紧的当且仅当$A(X)$的每个真理想是伪不动点.
英文摘要：
For a nonempty topological space X, the ring of all real-valued functions on $X$ with pointwise addition and multiplication is denoted by $F(X)$ and continuous members of $F(X)$ is denoted by $C(X)$. Let $A(X)$ be a subring of $F(X)$ and $B$ be a non-zero and nonempty subset of $A(X)$. Then we show that there are a subset $S$ of $X$ and a ring homomorphism $ϕ:A(X)\to A(S)$ such that $ker ϕ=Ann(B)$. A lattice ordered subring $A(X)$ of $F(X)$ is called $P$-convex if every prime ideal of $A(X)$ is an absolutely convex ideal in $A(X)$. Some properties of $P$-convex subrings of $F(X)$ are investigated. We show that the ring of Baire one functions on $X$ is $P$-convex. A proper ideal $I$ in $A(X)$ is called a pseudofixed ideal if $\bigcap \overline{Z[I]}\neq \emptyset $, where $ \overline{Z[I]}=\{cl_X f^{-1}(0) | f\in I\}$. Some characterizations of pseudofixed ideals in some subrings of $F(X)$ are given. Let $X$ be a completely regular Hausdorff space and let $A(X)$ be a subring of $F(X)$ such that $f \in A(X)$ is a unit of $A(X) $ if and only if $f^{-1}(0)=\emptyset$ and $C(X) \subseteq F(X)$. Then we show that $A(X)$ is a Gelfand ring and $X$ is compact if and only if every proper ideal of $A(X)$ is pseudofixed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
