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The aim of this article is to survey ring-theoretic properties of Kasch, theregularity and the injectivity of the ring of real-continuous functions on atopoframe $L_{ \tau}$, i.e., $\mathcal{R}L_\tau$. In order to study theseproperties, the concept of $P$-spaces and extremally disconnected spaces areextend to topoframes. For a $P$- topoframe $L_{ \tau}$, the ring$\mathcal{R}L_\tau$ is $\aleph_0$-Kasch ring. $P$- topoframes are characterizedin terms of ring-theoretic properties of the regularity and injectivity of thering of real-continuous functions on a topoframe. It follows from thesecharacterizations that the ring $\mathcal{R}L_\tau$ is regular if and only ifit is $\aleph_0$-selfinjective. For a completely regular topoframe $L_\tau$, weshow that $\mathcal{R}L_\tau$ is a Bear ring if and only if it is a $CS$-ringif and only if $L_\tau$ is extremally disconnected and also prove that it isselfinjective ring if and only if $L_{ \tau}$ is an extremally disconnected$P$-topoframe.

Some ring-theoretic properties of the ring of $\mathcal{R}L_\tau$