This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.The first part begins with the definition and elementary properties of the $\overline{\nu }$ order function associated to an ideal $I$ of a reduced analytic...

Rings of formal power series $k\left[\right[C\left]\right]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k\left[\right[C\left]\right]$ : for every $\sigma$ in $k\left[\right[C\left]\right]$ and $c$ in $C$, we let $v(c,\sigma )$ be the first element of the support of $\sigma$ which is greater than or equal to $c$. Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in $k\left[\right[C\left]\right]$. We prove that a cyclically valued ring is a subring of a power series ring $k\left[\right[C,\theta \left]\right]$ with...