I'm asked to prove this:
$X$ is an [$F$-space] if and only if every ideal is [absolutely convex]
I have a problem in proving the first direction that if $X$ is an $F$-space then every ideal is an absolutely convex one.
F-space: (the topological concept) F-space is a completely regular Hausdorff space for which every finitely generated ideal of the ring of real-valued continuous functions is principal, or equivalently every real-valued continuous function f can be written as $f = g |f|$ for some real-valued continuous function $g$.