ring of real-valued continuous functions

It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing $I$, denoted by $I^{sz}$ and $I_{sz}$, respectively. We study some properties of $I^{sz}$ and $I_{sz}$. Also, it is observed that the sum of any family of minimal prime ideals in the ring ${mathcal{R}} L$ is either ${mathcal{R}} L$ or a prime strongly $z$-ideal in ${mathcal{R}} L$. In particular, we show that the sum of two prime ideals in ${mathcal{R}} L$ such that are not a chain, is a prime strongly $z$-ideal.the formula is not displayed correctly!

Let RL be the ring of real-valued continuous functions on a frame L as the pointfree  version of C(X), the ring of all real-valued continuous functions on a topological space X. Since Cc(X) is the largest subring of C(X) whose elements have countable image, this motivates us to present the pointfree  version of Cc(X).
The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in RL. In particular, we will introduce the pointfree version of the ring Cc(X). We define a relation from RL into the power set of R, namely overlap. Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as Im(f)⊆S for every continuous function f:X→R and S⊆R.

Let $L$ be a completely regular frame and $\mathcal{R}L$ be the ring of real-valued continuous functions on $L$.
We consider the set $$\mathcal{R}_{\infty}L = \{\varphi \in \mathcal{R} L : \uparrow \varphi( \dfrac{-1}{n}, \dfrac{1}{n}) \mbox{ is a compact frame for any $n \in \mathbb{N}$}\}.$$ Suppose that $C_{\infty} (X)$ is the family of all functions $f \in C(X)$ for which the set $\{x \in X: |f(x)|\geq \dfrac{1}{n} \}$ is compact, for every $n \in \mathbb{N}$.
Kohls has shown that $C_{\infty} (X)$ is precisely the intersection of all the free maximal ideals of $C^{*}(X)$.
The aim of this paper is to extend this result to the real continuous functions on a frame and hence we show that $\mathcal{R}_{\infty}L$ is precisely the intersection of all the free maximal ideals of $\mathcal R^{*}L$.
This result is used to characterize the maximal ideals in $\mathcal{R}_{\infty}L$.