Abstract. We study a class of direct systems of rings satisfying a lifting property $(L)$ in order to generalize some properties known in $R[\infty ]$, $R(\infty )$ and $R\langle\infty \rangle $. Moreover, the following theorem is given, generalizing that $R\langle\infty \rangle $ and $R(\infty )$ are stably strong $S$ if $R$ has a finite valuative dimension. If $A=\lim_\to(S_j^{- 1}R[\Lambda_j],f_{kj})$ is a locally finite-dimensionsal domain, $f_{kj}$ are $R$-homomorphisms, and t.d.$[A:R]=\infty $, then $A$ is a stably strong $S$-domain. Finally, we present another characterization of rings satisfying the valuative altitude formula.

AMS Subject Classification (1991): 13C05, 13F05, 13F20

Received April 29, 1999, and in revised form March 2, 2000. (Registered under 2747/2009.)