Experience shows that in geometric situations the separating ideal associated with two orderings of a ring measures the degree of tangency of the corresponding ultrafilters of semialgebraic sets. A related notion of separating ideals is introduced for pairs of valuations of a ring. The comparison of both types of separating ideals helps to understand how a point on a surface is approached by different half-branches of curves.

In analogy with the notion of the composite semi-valuations, we define the composite $G$-valuation $v$ from two other $G$-valuations $w$ and $u$. We consider a lexicographically exact sequence $(a,\beta ):A_u\to B_v\to C_w$ and the composite $G$-valuation $v$ of a field $K$ with value group $B_v$. If the assigned to $v$ set $R_v=\{x\in K/v\left(x\right)\ge 0$ or $v\left(x\right)$ non comparable to $0\}$ is a local ring, then a $G$-valuation $w$ of $K$ into $C_w$ is defined with its assigned set $R_w$ a local ring, as well as another $G$-valuation $u$ of a residue field is defined with $G$-value group $A_u$.

We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...