Add to ORKGsummary:This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $\operatorname{dim}(L)$ is the supremum of the lengths $k$ of sequences $p_0< p_1< \cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily compact. This result applies to the frame $\Cal C_z(X)$ of all $z$-ideals of $C(X)$, provided the underlying Tychonoff space $X$ is Lindelöf. If the space $X$ is compact, then it is ...