Common properties of some function rings on a topological space
Abstract
For a nonempty topological space X, the ring of all realvalued 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 nonzero 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.02110
 Bibcode:
 2021arXiv210702110A
 Keywords:

 Mathematics  General Topology;
 26A21 (Primary) 54C40;
 13C99 (Secondary);
 A.0
 EPrint:
 17 pages