Given a sequence $M={M_n}_{n\in\bkN}$ of real positive numberslogarithmically convex, we study subrings $\Gamma_M$ of the ringof formal power series in $s$ variables whose coefficientssatisfy growth conditions with respect to $M.$ Under very few restrictiveconditions on $M,$ we get in these rings composition theorems.We study the following problem. Given $F$ in $(\Gamma_M)^{s},$ if${\cal A}\circ F$ belongs to $\Gamma_M,$ which $\Gamma_N$ does theseries ${\cal A}$ belong to ?We also prove that, given a good order on $\bkN^{s},$ we can divide any seriesin $\Gamma_M$ by a finite family of series $f_1,\dots,f_p$ in such a way thatthe quotients and the remainder belong to $\Gamma_M.$ Then wediscuss algebraic properties of $\Gamma_M$ as division propertiesmodulo an ideal, noetherianity and flatness. We get preparation theoremsof Malgrange type in these rings. We also prove a version ofArtin's theorem.
