Common properties of some function rings on a topological space

Abstract

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 $\phi:A(X)\to A(S)$ such that $ker \phi =Ann(B)$. 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.