$Z_c$-ideals and Prime ideals in the ring $\mathcal{R}_c L$

Abstract

We know that $C_c(X)$ is the largest subalgebra of $C(X)$ consisting of elements with the countable image.

The ring $\mathcal{R}_c L$ is an sub-$f$-ring of $\mathcal{R} L$ which its elements have the pointfree countable image

as the pointfree version of $C_c(X)$.

In the present paper, we introduce and investigate $z_c$-ideals in $\mathcal{R}_c L$.

Also, we study relation between $z_c$-ideals and prime ideals in $\mathcal{R}_c L.$

We prove that for any frame $L$,

$C_c(X) \cong \mathcal R_c ( \mathfrak O X ) \cong \mathcal R_c \beta L \cong \mathcal R_c^* L $, where $\beta L \cong \mathfrak O X $, and as for this we conclude that if $\alpha , \beta \in \mathcal{R}_c L$, $|\alpha | \leq |\beta |^q$ for some $q > 1$, then $\alpha$ is a multiple of $\beta$ in $\mathcal{R}_c L$.

Also, we show that $I J=I \cap J$ whenever $I$ and $J$ are $z_c$-ideals.

Finally, we prove that $\mathcal{R}_c L$ is a Gelfand ring.